Irving Stringham 
 
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TV 
 
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NEWG0ME8 MATHEMATICAL SERIES 
 
 ELEMENTS 
 
 OF 
 
 ANALYTIC GEOMETRY 
 
 BY 
 
 SIMON NEWCOMB 
 
 Professor of Mathematics, U. 8. Naiyy 
 
 NEW YORK 
 HENRY HOLT AND COMPANY 
 
 1884 
 

 Copyright, 1884, 
 
 By 
 
 HENRY HOLT AND CO. 
 
PEEFACE. 
 
 The author has endeavored so to arrange the present work 
 that it shall be adapted both to those who do and those who 
 do not desire to make a special study of advanced mathema- 
 tics. Believing it better that a student should learn a little 
 thoroughly and understandingly than that he should go over 
 many subjects without mastering them, the work is so con- 
 structed as to offer a wide range of choice in the course to be 
 selected. 
 
 The opening chapter contains a summary of the new ideas 
 associated with the use of algebraic language, which the stu- 
 dent is now first to encounter. His subsequent progress will 
 depend very largely on the ease and thoroughness with which 
 he can master this chapter. 
 
 The next seven chapters correspond closely to the usual 
 college course in plane analytic geometry; but the second sec- 
 tions of Chapters III. and IV. , as well as some sections of Chap- 
 ter VIII. , may be regarded as extras in this course. 
 
 If to this be added the part on geometry of three dimen- 
 sions, we shall have a course for those who expect to apply the 
 subject to practical problems in engineering and mechanics. 
 
 The second sections of Chapters III. and IV., together 
 with Part III., form an introduction to the modern projective 
 geometry; a subject whose elegance especially commends it to 
 the student of mathematical taste. The author has tried to 
 develop it in so elementary a way that it shall offer no diffi- 
 culty to a student who has been able to master elementary 
 geometry and trigonometry. 
 
 800557 
 
TABLE OF CONTENTS. 
 
 PART 1. PLANE ANALYTIC GEOMETRY. 
 
 Chapter I. Fundamental Conceptions m Algebra and 
 
 Geometry Page 3 
 
 Algebraic Conceptions, 3. Roots of Quadratic Equations, 7. 
 Proportional Quantities, 8. Geometric Conceptions, 11. 
 
 Chapter II. Co-ordinates and Loci 13 
 
 Cartesian or Bilinear Co-ordinates, 13. Problems, 15. Area 
 of Triangle, 20. Division of a Finite Line, 21. Polar Co-ordi- 
 nates, 23. Transformation of Co-ordinates, 25. Loci, 29. 
 
 Chapter III. The Straight Line 35 
 
 Section I. Elementary Theory. Equation of a Straight 
 Line, 35. General Equation of the First Degree, 38. Forms 
 of the General Equation, 39. Special Cases of Straight Lines, 
 42. Problems, 43. Normal Form, 46. Lines Determined by 
 given Conditions, 47. Relation of Two Lines, 51. Transforma- 
 tion to New Axis of Co-ordinates, 56. 
 
 Section II. Use of the Abbreviated Notation. Functions of the 
 Co-ordinates, 58. Theorems of the Intersection of Lines, 62. 
 
 Chapter IV. The Circle 73 
 
 Section T. Elementary Theory. Equation of the Circle, 73. 
 Intersection of Circles, 76. Polar Equation, 78. Tangents and 
 Normal, 78. Systems of Circles, 88. Imaginary Points of In- 
 tersection, 91. 
 
 Section II. Synthetic Oeometry of the Circle. Poles and 
 Polars, 94. Centres of Similitude, 98. The Radical Axis, 103. 
 Systems of Circles, 105. Tangent Circles, 106. 
 
 Chapter V. The Parabola 113 
 
 Equation of the Parabola, 113. Polar Equation, 115. Diame- 
 ters, 116. Tangents and Normals, 117. Equations referred to 
 Diameter and Tangent, 123. Poles and Polars, 125. 
 
Vi TABLE OF CONTENTS. 
 
 Chapter VI. The Ellipse 131 
 
 Equations aud Fundamental Properties, 131. Polar Equa- 
 tion, 135. Diameters, 138. Conjugate Diameters, 140. Equa- 
 tion referred to Conjugate Diameters, 144. Supplemental 
 Chords, 145. Relation of the Ellipse and Circle, 146. Area of 
 the Ellipse, 147. Tangents and Normals, 148. Reciprocal 
 Polar Relations, 157. Focus and Directrix, 161. 
 
 Chapter VII. The Hyperbola 107 
 
 Equation and Fundamental Properties, 167. Equilateral 
 Hyperbola, 170. Conjugate Hyperbola, 173. Polar Equa- 
 tion, 174. Diameters, 177. Conjugate Diameters, 179. Equa- 
 tion referred to Conjugate Diameters, 183. Tangents and 
 Normals, 184. Poles and Polars, 189. Focus and Directrix, 
 • 191. Asymptotes, 192. 
 
 Chapter VIII. General Equation of the Second Degree. 201 
 Fundamental Properties, 201. Change of Direction of Axes, 
 203. Classification of Loci, 205. The Parabola, 206. The 
 Pair of Straight Lines, 208. Summary of Conclusions, 210. 
 Similar Conies, 212. Families of Conies, 214. Focus and 
 Directrix, 217. 
 
 PART II. GEOMETRY OF THREE DIMENSIONS. 
 
 Chapter I. Position and Direction in Space 223 
 
 Directions and Angles in Space, 223. Projections of Lines, 
 224. Co-ordinate Axes and Planes, 225. Distance and Direc- 
 tion between Points, 230. Direction-Cosines, 234. Trans- 
 formation of Co-ordinates, 237. Polar Co-ordinates in Space, 240. 
 
 Chapter II. The Plane. ,... , 245 
 
 Loci of Equations, 245. Equation of the Plane, 246. Gen- 
 eral Equation of the First Degree, 247. Relations of Two or 
 More Planes, 254. 
 
 Chapter III. The Straight Line in Space 261 
 
 .♦ Equations of a Straight Line, 261. Symmetrical Equations, 
 263. Direction- V'ectors, 265. Common Perpendicular to Two 
 Lines, 267. Intersection of Line and Plane, 271. 
 
 Chapter IV. Quadric Surfaces 275 
 
 General Properties of Quadrics, 275. Centre and Diameter, 
 377. Conjugate Axes and Planes, 279. Diametral Planes, 280. 
 
TABLE OF CONTENTS. Vll 
 
 Principal Axes, 282. The Three Classes of Quadrics, 283. The 
 Ellipsoid, 283. The Hyperboloid of One Nappe, 284. The 
 Hyperboloid of Two Nappes, 286. Tangent Lines and Planes, 
 287. Generating Lines of the Hyperboloid of One Nappe, 289. 
 Poles and Polar Planes, 294. Special Forms of Quadrics, 298. 
 The Paraboloid, 298. The Cone, 299. The Pair of Planes, 
 300. Surfaces of Revolution, 300. 
 
 PART III. INTRODUCTION TO MODERN GEOMETRY. . . 305 
 
 The Principle of Duality, 305. The Distance-Ratio, 308. 
 The Sine-Ratio, 311. Theorems involving Distance- and Sine- 
 Ratios, 316. The Anharmonic Ratio, 321. Permutation of 
 Points, 323. Anharmonic Ratio of a Pencil of Lines, 326. An- 
 harmonic Properties, 327. Projective Properties of Figures, 332. 
 Harmonic Points and Pencils, 334. Anharmonic Properties of 
 Conies, 338. Pascal's Theorem and its Correlative, 341. 
 Trilinear Co-ordinates, 343. Line-Coordinates, 348. 
 
ANALYTIC GEOMETRY. 
 
 PART I. 
 PLANE ANALYTIC GEOMETRY. 
 
 CHAPTER I. 
 
 FUNDAMENTAL CONCEPTIONS IN ALGEBRA AND 
 GEOMETRY. 
 
 1. Analytic Geometry is a branch of mathematics in which 
 position is defined by means of algebraic quantities. 
 
 As an example of how position may be defined by quanti- 
 ties we may take latitude and longitude. The statement 
 
 "This ship is in lat. 47° N. and long. 52° W.'' 
 indicates to the expert a certain definite point on the earth's 
 surface near Newfoundland. 
 
 47° and 52° are the quantities indicating the position. 
 
 Algebraic Conceptions. 
 
 The following algebraic conceptions and principles should 
 be well understood by the student of Analytic Geometry. 
 
 2. Principles of Algehraic Language. 
 
 I. When an algebraic symbol is used in a statement, the 
 statement is considered true for any value of that symbol, 
 unless some limitation is placed upon it. 
 
 II. Any algebraic expression represents a quantity^ and 
 may itself be represented by a single letter. 
 
4 PLANE ANALYTIC GEOMETRY. 
 
 3. Constants and Variables. A quantity is called 
 Constant when a definite fixed value is supposed to be 
 
 assigned it; 
 
 Variable when, no definite value being assigned it, it is 
 subject to change. 
 
 We may, when we please, assign a definite value to a 
 variable. It then becomes, for the time being, a constant. 
 
 Example. We may regard the expression 
 x^ — a^x 
 
 as one in which x may take all possible values, while a remains 
 constant: x is then a variable. 
 
 But we may also inquire what definite value x must have 
 in order that the expression may vanish. We readily find 
 these values to be 
 
 a; = 0; x — a\ x = — a. 
 
 The quantity x then becomes a constant. 
 
 Again, we may think of a constant as undergoing variation: 
 it then becomes a variable. 
 
 Remark. The distinction of constant from variable is 
 not an absolute but only a relative one; that is, relative to 
 otlier quantities, or to our way of thinking at the moment. 
 The only absolute constants are arithmetical numbers. 
 
 4. Functions. When two variables are so related that a 
 change in one produces a change in the other, the latter is 
 called a function of the other. 
 
 Such relations between quantities are expressed by alge- 
 braic equations. 
 
 Example. In the equation 
 
 y = ax^h, 
 a change in a, 5 or x produces a change in y. Hence y is a 
 function of these quantities. 
 
 The value of an algebraic expression containing any sym- 
 bol will generally vary with that symbol. Hence 
 
 Any algehraic expression containing a variable is a func- 
 tion of that variable. 
 
FUNDAMENTAL CONCEPTIONS. 5 
 
 IndeiKiident Variables. When a quantity, y, is a func- 
 tion of another quantity, x, we may assign to x all possible 
 values, and study the corresponding values of y. 
 
 The quantity x is then called an independent variable. 
 
 5. Identical and Conditional Equatio7is. An equation 
 between algebraic symbols may be either 
 
 necessarily true, whatever values be assigned the symbols; 
 
 or true only when some relation exists between those 
 values. 
 
 An equation necessarily true is called an identical equa- 
 tion, or an identity. An equation only conditionally true 
 is called an equation of condition, or a condition. 
 
 Between the two members of an identity the sign = is 
 used; between those of a condition, the sign =. 
 
 Example. We have 
 
 {x -\- a) {x — a) -\- a^^ x^; 
 
 because the two members are necessarily equal for all values 
 of X and a. But the statement 
 
 ax — by = 
 
 can be true only when 
 
 b 
 
 a ^' 
 and is therefore a condition. 
 
 The question whether an equation is an identity or a con- 
 dition is settled by reducing or solving it. 
 
 If an identity, the two members may be reduced to the 
 same expressions, or, if we try to solve it, we shall only bring 
 out = 0. 
 
 If a condition, a value of x in terms of the remaining 
 quantities will be possible. 
 
 Theorem. An equation of condition becomes an identity 
 by solving it with respect to any one symbol, and suhstituting 
 the value of the symbol thus found in the equation. 
 
 Example. If in the ijreceding equation 
 ax — by = 
 
6 PLANE ANALYTIC GEOMETRY. 
 
 we substitute the value of x derived from it, we have 
 
 a-y- by = 0, 
 
 an identity. 
 
 Hence any eq^iatmi may he regarded as an identity hy 
 siqyj^osing any one of its symbols to represent that function of 
 its other quantities obtained by solving it. 
 Example. The equation 
 
 aP -by = (^ 
 changes into the identity 
 
 aP-by = 
 
 when we suppose P ^—y, 
 
 6. The symbol = is also used as the symbol of definition 
 when, in accordance with § 2, II., we use a symbol to repre- 
 sent an expression. For example, 
 
 ax -{- by ~ X 
 means, 
 
 *^ we use Xfor brevity, to represent the expression ax -\- hyJ^ 
 When the sign = follows an expression in this way, it may 
 be read, '^ which let us caliy 
 
 7. Lemma. Between the variables x arid y and the con- 
 stants A, B and C the identity 
 
 Ax-]- By -{- C=Q (1) 
 
 subsists when, and only vjhen, 
 
 ^ = 0, ^ = 0, C = 0. (2) 
 
 Proof. That the identity subsists in the case supposed 
 is obvious; that it subsists only in this case is seen by showing, 
 first, that if G were different from zero, the identity would 
 fail for a; = 0, ?/ = 0; and next, that, C being zero, the 
 identity would fail for a: = when B was finite, and for 
 y := when A was finite. 
 
 Kemark 1. The deduction of (2) from (1) rests on the 
 assumption that x and y are independent variables. If A and 
 B were regarded as variables, the conclusions would be 
 X = 0\ y = 0; (7=0. 
 
FUNDAMENTAL CONCEPTIONS. 7 
 
 Eemark 2. Note the great difference between the inter- 
 pretation of the equation 
 
 ax -\- hy -{- c = 
 
 and of the identity 
 
 ax -\- hy -\- c = 0. 
 
 The equation expresses a certain relation between the vari- 
 ables X and y such that to each definite value of x corresponds 
 a definite value of y^ namely, the value 
 
 ax + c 
 
 y = — r- 
 
 The identity expresses no relation between the quantities, 
 but requires zero values of a, h and c, 
 
 8. Roots of Quadratic Equations. Every quadratic equa- 
 tion is considered to have two roots, which may \)Q real and 
 unequaly real and equal, or imaginary. If the equation is 
 
 ax^ -f J^ -f c = 0, (a) 
 
 then, since we know the roots to be given by the equation 
 
 -I ±^V - 4:ac 2c 
 
 X = — 
 
 2a _ ^> ip |/J2 _ ^ac 
 
 we see that the roots will be 
 
 real when b^ — ^ac > 0, i.e., when h^ — 4ac is positive; 
 
 real and equal when H^ — ^ac = 0; 
 
 imaginary when V^ — 4ac < 0; i.e., when Z>^ — ^ac is 
 negative. 
 
 The student should now be able to explain the following 
 special cases: 
 
 1. If the absolute term c vanishes, the roots become 
 
 X = and x = . 
 
 a 
 
 2. If J and c both vanish, both roots become zero. 
 
 3. If a approaches zero as its limit, one root increases 
 
 without limit, and the other approaches the limit — j. 
 
8 PLANE ANALYTIC GEOMETRY. 
 
 Hence we may say: When the coefficient of x^ in the quad- 
 ratic equation (a) vanishes, the two roots are 
 
 a: = — T and a: = oo . 
 
 
 
 4. If both a and b vanish while c remains finite, both 
 roots increase to infinity. 
 
 5. If «, h and c all vanish, the roots are entirely indeter- 
 minate, and the equation is satisfied by all values of x, 
 
 9. Froportio7ial Quantities. The quantities of one series, 
 a, h, c, etc., are said to be proportional to those of another 
 series. A, B, C, etc., when each quantity of the one series is 
 equal to the corresponding quantity of the other multiplied 
 by the same factor. 
 
 The fact of such proportionality is expressed in the vari- 
 ous forms: 
 
 a : A = b : B — c : G = etc.; 
 
 a :h : c '. etc. = A : B : C : etc.; 
 
 a = pA, b = pB, c = pC, etc.; 
 p being, in the last case, the multiplying factor. 
 
 TEST EXERCISES* 
 
 1. A point at the distance (1) from one side of a right 
 angle and at the distance (2) from the other side will be at 
 the distance (3) from the vertex of the angle. 
 
 Here the student will substitute symbols at pleasure for 
 (1) and (2), and will replace (3) by the proper function of 
 those symbols, reading the statement accordingly. For ex- 
 ample, he may put 
 
 X in place of (1) and y in place of (2), 
 
 y -{- xm place of (1) and y — x'ln place of (2), 
 etc. etc. etc., 
 
 * These exercises are designed to decide the question whether the 
 student has a suflScieut command of algebraic language and of geometri- 
 cal conceptions to enable him to proceed with advantage to the study 
 of Analytic Geometry. If he can perform all the exercises with ease, 
 he is probably well prepared to go on ; if he performs them only with 
 diflflculty, he may need much assistance in understanding the subject. 
 
FUNDAMENTAL (JONGEPTIONIS. 9 
 
 and in each case he must read the statement witli the j^ropcr 
 expression in place of (3). 
 
 2. If, in the preceding example, (1) varies while (2) re- 
 mains constant, the point will describe a line to 
 
 Fill the blanks with appropriate words. 
 
 3. If (2) varies while (1) remains constant, the point will 
 describe a line to the . 
 
 4. If (1) remains equal to (2), but both vary, tlie point 
 will move along the . 
 
 5. If (1) and (2) vary in such a way that (3) remains con- 
 stant, the point will describe a of radius around 
 
 as a — . 
 
 6. If two fixed points, A and B, are at the distances from 
 each other, and if a third point, P, be taken at the distance 
 (4) from each of these points; then, if (4) varies, the point P 
 will describe a line {define the situation of tlie Ime). But, in 
 varying, the distance (4) cannot become less than — . 
 
 The numbers in parentheses are to be replaced by appro- 
 priate symbols or expressions. 
 
 7. If, in the preceding example, a point be taken at the 
 distance (5) from the point ^4, and at the distance (6) from B-, 
 then, if (5) varies while (6) remains constant, the point will 
 describe a of around as a — . 
 
 8. But if (6) varies while (5) remains constant, the point 
 will 
 
 9. If the constant value of (6) in Ex. 7 plus the con- 
 stant of (5) in Ex. 8 = a, the two will be to each 
 
 other. 
 
 10. If a line be drawn so as to pass at the distance r 
 from each of the preceding points, then, if r varies, the 
 line will turn. round {describe liow it will turn). But the 
 
 value of r can never exceed , and for each value of r 
 
 there will be two positions of the line making equal angles 
 with — . 
 
 11. If the line be required to pass at the distance (7) from 
 the point A, and at the distance (8) from the point B\ then, 
 if (7) varies while (8) remains constant, the line v/ill move 
 
10 PLANE ANALYTIC OEOMETRT. 
 
 round so as always to be tangent to the around — as 
 
 a with radius — . 
 
 12. What symbols must (9) and (10) be replaced by in 
 order that all values of x and y which satisfy the equation 
 
 Ax-\- By -\- G =0 
 
 may also satisfy the equation 
 
 mAx + (9)?/ + (10) = 0; 
 
 that is, in order that these two equations may give the same 
 value of y in terms of a;? 
 
 13. Show that the identity 
 
 ax -\- hy -\- c ^ Ax -^ By -\- G, 
 X and y being variables, is impossible unless 
 a =A; b = B; c = G. 
 
 14. If we put 
 
 F = x - 2y -}-3c, P' = 3x - 6y + 9c, 
 is it possible to form an identity of the form 
 P + mP' = 0, 
 
 and, if so, what will be the value of ?n? 
 
 15. Generalize the preceding result by showing that if we 
 have 
 
 f P = ax-\-by -j- c, P' = Ax -^ By ~\- G, 
 
 the identity P + mP' = 0, 
 
 X and y being variables, is possible only when 
 
 a \ A = l : B = c I G, 
 and express the value of m, 
 
 16. 11 a -\- X remains constant, and x varies at the rate of 
 plus one foot a second, at what rate will a vary? 
 
 17. What will be the answer to this last example if it is 
 (I — 2x instead of a -\- x which remains constant? 
 
 18. If X may take any values between the extremes — 1 
 
 and + 2, between what extremes will the value of — '—z, be 
 
 X — 1 
 
 contained? 
 
FUNDAMENTAL CONCEPTIONS. H 
 
 Geometric Conceptions. 
 
 10. A geometric concept, form, or figure of any kind, may 
 be called a geometric object.* 
 
 In the higher geometry all geometric objects, when not 
 qualified, are considered as complete in every particular. 
 
 Examples. A straight line is considered to extend to 
 infinity in both directions. When a terminating straight line 
 is treated, it is considered as that portion of an infinite straight 
 line contained between some two points. 
 
 A triangle is considered as formed by three indefinite 
 straight lines intersecting each other in three different points. 
 
 Geometric objects differ from each other in magnitude, 
 form and situation. 
 
 Points, straight lines and planes can, however, differ only 
 in situation, because any two points, any two lines or any two 
 planes may be made to coincide with each other by a change 
 of situation. 
 
 11. Points at Infinity. A pair of parallel lines are said to 
 intersect in ?i point at infinity; that is, in a point at an infinite 
 distance. 
 
 The idea of a point at infinity is reached in this way: Let 
 us suppose one of two intersecting lines to turn round on one 
 of its points and gradually to approach the position of parallel- 
 ism to the other line. As this position is approached the 
 point of intersection of the two lines will recede indefinitely, in 
 such wise that while the revolving line approaches parallelism 
 as its limit, the point will recede beyond every assignable limit. 
 
 Conversely, if we suppose the point of intersection to recede 
 indefinitely along the fixed line, the moving line will approach 
 
 * This is the best English word which has presented itself to the 
 author to correspond to the Gebild of the Germans. Such a word is needed 
 in the higher geometry as a term of the most general kind to express 
 the things reasoned about. The term magnitude is too limited, not only 
 because a point is to be included among geometric objects, but because 
 objects are considered not merely as magnitudes but, in a more general 
 way, as things of which magnitude is only one of the qualities. 
 
12 PLANE ANALYTIC GEOMETRY. 
 
 the position of parallelism as its limit. This limit will be 
 the same whether the point of intersection recedes in one 
 direction or in the opposite. 
 
 Hence, using the convenient language of infinity, we see 
 that when the point of intersection is at infinity in either 
 direction the two lines are parallel. There is, therefore, no 
 need of making any distinction between these supposed points, 
 and they are talked about as a single point, called the point at 
 infinity. 
 
 The principle here involved is of extensive application in 
 the higher mathematics, and may be expressed thus: , 
 
 Instead of ^ismg new or different forms of language to 
 meet excej^tional cases, we use the common language, hut put an 
 exce])tional interpretatioyi ujjon it. 
 
 The advantage of this way of speaking is that we are not obliged to 
 make any exceptional cases respecting the intersection of lines when the 
 two lines become parallel. 
 
 The proposition, Two straight lines intersect in a single point, is then 
 considered universally true, the point being at infinity when the lines 
 are parallel. 
 
 The following is a convenvient illustration of this form of language. 
 Let it be required to draw a line through a fixed point, P, so as to inter- 
 sect the fixed line h at the ^ 
 
 same point, Q, where the line ~h ^„^--^^ 
 
 a intersects it. The construc- 
 tion will be literally possible so 
 long as a and b intersect, but 
 will cease to be literally possible ^ 
 if a takes the position a', paral- 
 lel to b, because then there will be no point Q of intersection. 
 
 But let us interpret the problem in this way: The required line must 
 intersect b where a intersects b. But in case of parallelism, a intersects 
 b nowhere. Hence the required line must intersect b nowhere; that is, 
 it must be parallel to it. It is this particular noiohere which is called the 
 point at infinity on the line b. It is, moreover, clear that if Q recedes to 
 infinity, both the line a and the required line will approach the position 
 of parallelism to b as their respective limits. 
 
CHAPTER II. 
 
 OF CO-ORDINATES AND LOCI. ' 
 
 12. Def. The co-ordinates of a geometric object are 
 those quantities whicli determine its situation. 
 
 Co-ordinates, like other quantities, are represented by 
 numerical or algebraic symbols. 
 
 The situation of an object is defined by its relations 
 to some system of points or lines supposed to be fixed. Such 
 a system is called a system of co-ordinates. 
 
 There are several systems of co-ordinates to be separately 
 defined. 
 
 First System : Cartesian or Bilinear 
 Co-ordinates. 
 
 13. On this system the position of a point is fixed by its 
 relation to two intersecting straight lines called axes. 
 
 Let AX and BY \iQ the two lines, and their point of 
 intersection. -y- 
 
 The point is then / 
 
 called the origin. / 
 
 The indefinite line y- yP 
 
 AX, which we may con- / 
 
 ceive to be horizontal, is / 
 
 called the axis of ab- A '^ ~ 
 
 scissas, or the axis of X. / 
 
 The intersecting line / 
 
 BY'i^ called the axis of / 
 
 ordinates, or the axis ^' 
 
 of r. 
 
 Let P be the point whose position is to be defined. 
 
14 PLANE ANALYTIC GEOMETRY. 
 
 From P draw PM parallel to OY and meeting the axis 
 of X in Mj and 
 
 PiV parallel to OX and ^ 
 meeting the axis of J^in / 
 N. n/ ,p 
 
 Then either of the / 
 
 equal lines OM, NP is / 
 
 called the abscissa of the -^ J. Z X 
 
 point P; I J&. 
 
 Either of the equal / 
 
 lines MP, ON is called / 
 
 the ordinate of ■ the -g/ 
 
 point P. 
 
 It is evident that for every position we assign to P the 
 abscissa and ordinate will each have a definite value. 
 
 14. Co-ordinates Detennine a Point. When the lengths 
 OM and MP are given, the point P is completely determined 
 in the following way: We measure from on the axis of X 
 the given distance OM. 
 
 Through M we draw an indefinite line parallel to the axis 
 of Y, and on this line measure a length MP. 
 
 The single point P which we thus reach is the point which 
 has the given abscissa and ordinate. 
 
 Because the abscissa and ordinate thus determine the 
 situation of P, they form, by definition, a pair of co-ordi7iates 
 ofP(§12). 
 
 Notation. The abscissa is represented by the symbol x. 
 The ordinate is represented by the symbol y. 
 
 It is evident that if the point P be fixed in position, its 
 co-ordinates will be constants. But if P varies, one or both 
 of the co-ordinates will vary also. 
 
 15. Algelraic Signs of the Co-ordinates. In what pre- 
 cedes it is supposed that the direction, as well as the distance, 
 of the measures OM and MP is given. If these directions 
 were arbitrary, we might measure the given distance OM in 
 either direction from 0, and thus reach either the point Mio 
 the right of or the point M' to the left of 0. 
 
CO-ORDINATES AND LOCI. 
 
 15 
 
 By measuring the ordinate in either direction from the 
 points M and M' we sliould reach cither of four points, 
 P, P', P", jP'", of which the co-ordinates would all be equal 
 iw absolute value. 
 
 To avoid ambiguity in this respect the algehraic sign 
 of the abscissa is supposed 
 
 positive when meas- 
 ured from towards the 
 right, and 
 
 negative when meas- 
 ured towards the left. 
 
 The ordinate MP is 
 supposed 
 
 positive when meas- 
 ured upiDard, and 
 
 negative when meas- 
 ured doivnward. 
 
 Now if the abscissa 
 X = OM = a and the 
 ordinate y = MP = d, then the 
 
 co-ordinates of P are x = -[- a, 
 co-ordinates of P' are x = — a, 
 co-ordinates of P" are x = ^ a, 
 co-ordinates of P'" are x =^ -\- a. 
 
 y = + ^; 
 y = - h 
 y = — b. 
 
 Thus the ambiguity is completely avoided when the algebraic 
 signs of the co-ordinates, as well as their absolute values, are 
 given, so that only one point corresponds to one pair of alge- 
 braic values of the co-ordinates. 
 
 16. Rectangular Co-ordinates. When not otherwise ex- 
 pressed, the axes of co-ordinates are supposed to intersect at 
 right angles. The co-ordinates are then called rectangular 
 co-ordinates. 
 
 To designate a point by its co-ordinates we enclose the 
 symbols or numbers expressing the co-ordinates between pa- 
 rentheses, with a comma between them, writing the value of 
 X first. 
 
16 PLANE ANALYTIC GEOMETRY. 
 
 Example. By (2, 3) we mean ^*the point of which the 
 abscissa is 2 and the ordinate is 3." 
 
 EXERCISES. 
 
 1. Draw a pair of rectangular axes, and, taking a centi- 
 metre or inch, as may be most convenient, for ihe unit, lay 
 down the position of points having the following co-ordinates: 
 
 (+ 2, + 3), (+ 2, - 3), (- 2, + 3), (- 2, - 3), 
 (+ 3, + 2), (+ 3, - 2), (- 3, + 2), (- 3, - 2). 
 
 Show that these eight points all lie on a circle having the 
 centre as its origin. What is the radius of this circle? 
 
 2. Mark a number of points of each of which the ordinate 
 shall be equal to the abscissa. How are these points situated? 
 
 3. Mark the points (1, - 1), (2, - 2), (- 1, 1) and 
 (— 2, 2), and show their relations. 
 
 4. Mark the points (1, 2), (2, 4), (3, G), (4, 8), and show 
 how they are situated relatively to each other. 
 
 5. If we join the points {a, — i) and {a, h) by a straight 
 line, what will be the direction of this line? 
 
 6. Find, in the same wa}-, the direction of the line joining 
 the points {a, h) and (— a, b); (a, h) and {— a, — h). 
 
 7. Show that the distance of the point {a, h) from the 
 origin is Va"" -f W. 
 
 8. If we mark all possible points for which y has the con- 
 stant value -f- 1, how will these points be situated? 
 
 1*7. Problem I. To express the distaiice between two 
 ^joints ivhose co-ordinates are giveyi. 
 
 When the co-ordinates of two points are given, the position 
 of each point is completely determined (§ 14). 
 
 Therefore the distance between the points is completely 
 determined, and may be measured geometrically. 
 
 The algebraic problem requires us to express this distance 
 algebraically in terms of those quantities which determine the 
 position of the points, namely, their co-ordinates. 
 
 In the figure let P' and Pbe the two points; x', y\ the co- 
 ordinates of P'; and x, y, the co-ordinates of F. 
 
CO-ORDINATES AND LOCI. 
 
 17 
 
 Then we shall have 
 
 OM' = x\ 
 P'M' == if, 
 
 OM = x\ 
 PM = y. 
 
 If from P' we drop 
 a perpendicular, P'R, 
 upon MP, we shall have, 
 from the right-angled 
 triangle P'PR, ^ 
 
 P'R = M'M =x - 
 
 and RP = MP - MR = y - 
 
 Then, by the Pythagorean proposition, 
 
 Y 
 
 
 
 p/ 
 
 
 
 
 
 } 
 
 / 
 
 R 
 
 
 
 
 
 
 
 b/ 
 
 / 
 
 
 
 X 
 
 
 / 
 
 j\: M 
 
 
 P'P' = P^R' + RP\ 
 
 Let us then put d = the distance P'P. 
 By substituting the values in terms of the co-ordinates and 
 extracting the square root, we shall have 
 
 d = ^/{{x - x'r + {,j - y'y\, (1) 
 
 which is the required expression for the distance of the points 
 in terms of their co-ordinates. 
 
 IS. Peoblem II. To express the ajigle which the line 
 joining two points, given hy their co-ordinates, malces with 
 the axis of X, 
 
 Using the same construction as before, let B be the point 
 in which the line PP' intersects the axis of X. 
 
 The required angle will then be 
 
 PBX or PP'R, 
 
 If we put 
 
 ■ f E the required angle, 
 
 we shall have, by trigonometry, 
 
 RP — P'P sin 6 = f7 sin f; 
 
 P'R = P'P cos e z=,d cos f ; 
 whence, by division, 
 
 RP y - ?/' 
 
 tan e 
 
 P'R 
 
 X — X' 
 
 (3) 
 
18 PLANE ANALYTIC OEOMETRT. 
 
 The last equation gives tlie required expression for the 
 tangent, from which e may be found. 
 
 19. The two preceding problems may be more elegantly 
 solved by a single pair of equations: 
 
 d sin € — y — 
 d cos e 
 
 ii~y';,) (3) 
 
 X — X . ) 
 
 The method of solving these equations is explained in 
 trigonometry. 
 
 20. Problem III. Ttvo points heing given hy their co- 
 ordinates, it is required to find the p)oints in ivhich the straight 
 line joining them intersects the respective axes of co-ordinates. 
 
 Solution. Let B be the point in which the line inter- 
 sects the axis of x, C the point in which it intersects the axis 
 oiy. 
 
 The point B will then be given by the value of OB, its 
 abscissa, which we denote by x^, and C by the value of 00, 
 its ordinate, which we denote by y^. 
 
 In the similar triangles MBP and RP'P we have 
 
 BM : MP = P'R : RP. 
 
 Substituting for the lines their values in terms of the co- 
 ordinates, this gives 
 
 y -y 
 
 whence 
 
 OB = OM - BM^x- ^i^Zli'i 
 
 y - y 
 
 or X = ^^^' ~ ^'^ ~ y^^ " ^'"^ = ^y' ~ ^''^ u\ 
 
 y -y' y' -y ' _ 
 
 The value oi 00 can be found by a similar construction, 
 but we may also deduce it from OB by the equation 
 
 00 = OB tan e. 
 
 But in the 'figure as drawn falls below 0, so that the 
 value oi 00 just obtained is the negative of the required 
 
C0-0BDINATE8 AND LOCI. 19 
 
 ordinate of the point of intersection. This co-ordijiate being 
 y^f we shall have 
 
 y,= -OB tan « = ^?^X (5) 
 
 The student should now note the relation between the 
 conditions of the geometric and the algebraic solutions. 
 The problem considered as a geometric one is: 
 
 Tivo points leing given in position^ to find the intersection 
 of the straight line joining them luith the axes of co-ordi7iates» 
 
 The problem is solved geometrically simply by drawing the 
 line. The algebraic requirement is: 
 
 Ttvo points being giveii hy rneaiis of their co-ordinates, it is 
 required to express the poitiis in which the straight line join- 
 ing them intersects the co-ordinate axes in terms of the respec- 
 tive co-ordi7iates of the given points. 
 
 The algebraic solution is given by the equations (4) and 
 (5). 
 
 21. The preceding problems illustrate the following gen- 
 eral principle: 
 
 Whenever one geometric object is determined by another 
 geometric object, the algebraic quantities which define the one 
 can be exjjressed in terms of those quantities ivhich define the 
 other. 
 
 EXERCISES. 
 
 1. Lay down the four points (1, 1), (1, 2), (2, 2), (2, 1), 
 and join each one and that next following so as to form a 
 quadrilateral. What will be the nature of this quadrilateral? 
 
 2. Show that the points (1, 0), (1, 1), (2, 0), (2, 1) lie 
 at the four vertices of a square. 
 
 3. Show that each of the following sets of four points 
 are the vertices of a parallelogram : 
 
 Set (a): (0, 0), (3, 1), (0, 4), (3, - 3)- 
 ' Set {b): (1, 3), (2, 5), (6, 5), (5, 3); 
 Set (c): (1,1), (2,4), (5,5), (4,2). 
 
 4. Show by a geometric construction, employing the 
 properties of similar triangles, that each of the lines joining 
 
PLANE ANALYTIC GEOMETRY. 
 
 the following pairs of points passes through the origin of co- 
 ordinates: 
 
 (a): a line joining points (1, 1) and (2, 2); 
 
 (b): a line joining points (1, 2) and (3, 6); 
 
 (c) : a line joining points (1, 3) and (— 1,-3); 
 
 (d): a line joining points (a, b) and (71a, nb). 
 
 Show in the same way that each of the following trip- 
 
 (3, 3); 
 (3, 4); 
 (1> + 4); 
 (- 1, 0); 
 
 (a -j- npf b -\- nq). 
 
 0. 
 lets of points lies in a straight line: 
 
 (a): (1,1), (2,2), 
 
 {b): (1,0), (2,2), 
 
 (6'): (-1,0), (0, +2), 
 (.0:(3, -2), (1,-1), 
 (c): {a, b), {a-\-p,b -\- q), 
 
 6. What are the distance and direction (relatively to the 
 axis of X) from the point (1, 2) to the point (4, G)? 
 
 23, Problem IV. To find the area of a triangle, the 
 co-ordinates of the vertices beiiig given, 
 
 Kemakk. Since the positions of the vertices completely 
 determine the triangle, and therefore determine its area also, 
 it follows from the general principle, § 21, that this area can 
 be algebraically expressed in terms of the co-ordinates of the 
 vertices. 
 
 Solution. Let P, P' and P" be 
 the vertices, and {x, y), {x% y') and 
 {x", y") their respective co-ordi- 
 nates. 
 
 Let us put A the area of the 
 triangle. We shall then have 
 
 A — area PMM"P^' plus area 
 M'P'P"M'' minus area MWP'P. 
 
 In these three trapezoids we have 
 area MPP"M" = i{3IP + iWP'')MiW 
 
 area J/"P"P'J/' =4(i¥"P" + M'P')M"M' 
 
 area MPP'M' = i(M'P' + MP)MM' 
 
 M 
 
 M" M' 
 
 - .-^0; 
 
 = W + y) (^' - ^). 
 
CO-ORDINATES AND LOCI. 
 
 21 
 
 Therefore 
 2J = (2/ + y") (X" 
 
 X) + iy" + y') (^' - 
 + (y'- 
 
 x") 
 
 [- y) (*■ - ^'), 
 
 ') + y"{x' - X), (6) 
 
 or, by reduction, 
 
 'ZJ = y(x" - x') + y\x ■ 
 which is the required expression. 
 
 33. To divide a finite line into segments having a given 
 ratio. A finite line is defined by the co-ordinates of its two 
 terminal points. Let us now consider the problem: 
 
 To find the co-ordinates of the poiiit luhich divides the finite 
 line joining two given poifits into seginents having a given 
 ratio. 
 
 Let us put: 
 
 x^y y^, the co-ordinates of one end, A, of the line. 
 x^, y^, the co-ordinates of the other end, B. 
 A, ^, the given ratio. 
 
 X, y, the co-ordinates of the required point, P. 
 Draw ^iVland PQ each 
 parallel to the axis of X, and 
 PM, BN each parallel to 
 the axis of Y. Then 
 AM = x — x^; PQ = x^—x; 
 PM=y-y^; BQ = y^-y. 
 Since we require that 
 
 AP : PB = ?i: M, 
 we have the proportion 
 
 X: /.i = AP : PB 
 
 = AM: PQ = X - X 
 = PM:BQ = y-y^ 
 We hence deduce the equations 
 
 X{x^ - x) = /j{x - .tJ, 
 
 Hy. - y) = M{y - y,), 
 
 which give 
 
 \x, -f fix^^ ^_ _ Ay, + fAy 
 
 : x^ 
 
 -y^ 
 
 — X 
 
 -y- 
 
 X = 
 
 y = 
 
 A + yU ' ^ A^-/^ 
 
 which are the required co-ordinates of the point of division. 
 
 (7) 
 
22 PLANE ANALYTIC OEOMETBT. 
 
 Corollary. If P is to be the middle point of the line, we 
 have A = yw, whence 
 
 or. 
 
 Each co-ordinate of the iniddle point of a line is half the 
 sum of the co7Tes2J07idi7i(/ co-ordinates of its terminal poi?its. 
 
 EXERCISES. 
 
 1. Express the co-ordinates of the middle point of the line 
 terminatiDg in the points (1, 6) and (3, — 4). 
 
 2. One end of a line is at the point {—2, — 3) and its 
 middle point at (1, — 2). Where is the other end? 
 
 3. Find the middle point of that segment of the line join- 
 ing the points (— 1, 6) and (3, — 2) which is contained be- 
 tween the axes of co-ordinates. A^is. (1, 2). 
 
 4. A line terminating at the points (1, 6) and (3, — 4) is 
 to be divided into four equal segments. Find the co-ordi- 
 nates of the three dividing points. 
 
 5. The line joining the points {a, h) and {p, q) is to be 
 divided mio five equal parts. Express the co-ordinates of the 
 four points of division. 
 
 6. What is the distance between the middle points of the 
 lines whose respective termini are in the points (1, 7), (— 5, 3) 
 and (0, 2), (6, - 4)? 
 
 7. What point bisects the line from the origin to the 
 middle point of the line terminating at the points (7, — 0) 
 and (- 3,-7)? 
 
 8. Find the co-ordinates of the point which is two thirds 
 of the way from the point {a, h) to the point {a' , h'). 
 
 9. Prove the theorem that the three medial lines of a tri- 
 angle meet in a point two thirds of the way from each vertex 
 to the opposite side, as follows: 
 
 Let {x^, yj, {x^, y^) and {x^, y^) be the three vertices of 
 the triangle. 
 
 Express the middle point of each side. 
 
 Then express the co-ordinates of those three points which 
 
CO-ORDINATES AND LOCI. 23 
 
 arc respectively two thirds of the way from the several vertices 
 to the middle points of the opposite sides, and thus show that 
 the three points are coincident. 
 
 10. Prove that the lines joining the middle points of the 
 opposite sides of a quadrilateral and the line joining the 
 middle points of the diagonals all bisect each other. 
 
 To do this, express the co-ordinates of the middle points of the sides 
 and of the diagonals, and then of the middle points of the three joining 
 lines, and show that the latter points are the same for each joining line. 
 The very simple proof of this theorem which is thus found affords a 
 striking example of the power of the analytic method. 
 
 Second System : Polar Co-ordinates. 
 
 The position of a point may be defined by its distance and 
 direction from a fixed point. 
 
 The fixed point is then called the origin. 
 
 The distance of the point from the origin is called the 
 radius vector of the point. 
 
 In plane geometry the direction of a point from the origin 
 is fixed by the angle which the radius vector makes with an 
 adopted base-line. 
 
 Let OX be the base-line, and P the point; P being in 
 any one of the positions P', P", etc. 
 
 OP will then be the radius vector, and the angle XOP 
 will be the required angle. 
 
24 
 
 PLANE ANALYTIC GEOMETRY. 
 
 We generally put 
 
 r = the radius vector OP, and 
 
 6 E the angle XOP, which is called the vectorial angle. 
 
 The former is always considered positive, being measured 
 from the origin, 0, in the direction OP. The latter is posi- 
 tive when measured in the direction opposite to that in which 
 the hands of a watch move, and negative in the opposite di- 
 rection, just as in trigonometry. 
 
 34. Problem. To express the distance hekvee^i two points 
 in terms of their polar co-ordinates. 
 
 Let P and Q be the points. 
 
 In the triangle P0§ we have, y 
 by trigonometry, 
 
 P§== r^ + r" - 2rr' cos POQ 
 =zr''-\-r''-2rr'co^{d-e'), 
 
 6 and 6' being the angles which 
 the radii vectores make with the 
 initial or base line; therefore 
 
 PQ = {r' + r'' - %rr' cos {0 
 which is the distance required. 
 
 0')!' 
 
 (9) 
 
 EXERCISES. 
 
 1. Show how a point will be situated when its vectorial 
 angle is in the first, second, third and fourth quadrant re- 
 spectively. 
 
 2. If the vectorial angle d' and radii vectores r and r* are 
 constant, while Q may vary at pleasure, for what values of d 
 will the distance of the points be the greatest and least i)Ossi- 
 ble, and what will be the greatest and least distances? Show 
 the correspondence of the algebraic answer from equation (9) 
 with the obvious answer from the figure. 
 
 3. If r = r' and 6 ■\- B' = 180°, show both geometrically 
 and algebraically that distance = 2r cos 0. 
 
 4. If ^ - 0' = 90° or 270°, express the distance of the 
 points both by a diagram and by the equation (0). 
 
CO-ORDINATES AND LOCI. 
 
 25 
 
 Transformation of Co-ordinates 
 System to Another. 
 
 from One 
 
 The general problem of the transformation of co-ordi- 
 nates is this: 
 
 Given: 1. The co-ordinates x and y of a point P referred 
 to some system of co-ordinates. 
 
 Given: 2. The position of a second system of co-ordinates 
 m relation to the other syste?)i. 
 
 Kequired: To express the co-ordinates of P when referred 
 to the second system, 
 
 25. Relation of Rectangular and Polar Co-ordinates. 
 
 Let OX, OF be the rectangular axes, and P the position 
 of any point. 
 
 1st. We shall suppose the origin to be taken as the pole, 
 and the axis of abscissas as the base or initial line; then we 
 shall evidently have Y 
 
 X = r cos 
 and y = r sin 0. 
 
 To express the poiar co-ordi- 
 nates in terms of the rectangular ^ 
 co-ordinates, we have from the ' ^ M 
 
 last two equations, by squaring and adding, 
 r^ = x^ -\- y^, or 
 
 and, by division, tan Q = 
 
 = V^Tl/% 
 
 which determine r and 6 when x and y are given 
 
 2d. If the initial or base line 
 instead of coinciding with the 
 axis of X makes an angle a with 
 it, we shall evidently have, from 
 the figure, 
 
 X = r cos {a -f 6) 
 and y = r sin (a -\- 6), 
 
 whence r = Vx"" -f y"" and tan {a -J- 6) 
 
 
 0^ 
 
 
 x 
 
26 PLANE ANALYTIC OEOMETBY. 
 
 EXERCISES. 
 
 1. In the figure of § 24, express the area of the triangle 
 OPQ in terms of r, r' and 6 - 6'. 
 
 2. If r, r' and r" are the i^dii vectores, and 6, 6' and 6" 
 tlie corresponding angles of three points, which we shall call 
 P, F' , P" , it is required to express the areas, first, of the 
 triangles OPP', OP'P" and OPP", and then of PP'P'\ 
 
 3. The point (3, 3) is the centre of a circle of radius 2, in 
 which two diameters, each making angles of 45° with the axes, 
 are drawn. Find the polar co-ordinates of the ends of these 
 diameters. 
 
 4. The point (a, b) is the centre of a circle of radius P. 
 From the centre is drawn a radius making an angle y with 
 the axis of X. Express the rectangular co-ordinates of the 
 end of this radius. 
 
 26. Transformation from one rectaiigular system to 
 anotlier. 
 
 Solution, Let us first suppose 
 the two systems of co-ordinates par- N 
 allel. Also suppose 
 OXy OY the axes of the original 
 
 system ; 
 O'X', O'Y' the axes of the second 
 
 system; 
 P the point whose co-ordinates are x and y in the old system. 
 
 Draw 
 
 PM'MW Y0\\ Y'O', 
 PN'N\\XO\\X'0', 
 and put 
 
 a = the abscissa of the new origin, 0', referred to the 
 old system; 
 
 b = the ordinate of 0'; 
 
 x^ = the abscissa O'l/' of P referred to the new system; 
 
 y' = the ordinate M^P of P referred to the new system. 
 We then have 
 
 M 
 
 M x' 
 
 _X 
 
 x' - X - «; ) ^^^^ 
 
 y = y - h^ 
 
CO-ORDINATES AND LOCI. 
 
 27 
 
 which are the required expressions for the new co-ordinates in 
 terms of the old ones. 
 
 3*7. Secondly. Suppose the new axes to make an angle, 
 6, with the old ones, but to have the same origin. 
 
 Let us put 
 r E the radius vector 0P\ 
 q) E the angle XOP. 
 
 We shall then have 
 
 Angle X'OP = cp - 6, 
 
 Putting, as before, 
 x' and y' for the co-ordinates referred to the new system, and 
 X and y the co-ordinates of the old system, we have, by § 25, 
 
 X = r cos cp', 2/ = ^' sin cp', (a) 
 
 x' = r cos((p — S); y' = r s'm((p — 6). (d) 
 
 By trigonometry, 
 
 cos((p ■— 6) = cos q) cos d -|- pin q) sin S; 
 sin(<^ — 6) = sin cp cos S — cos cp sin d. 
 
 Substituting these values in (b) and eliminating r and cp 
 by («), we have 
 
 x^ = y sin 6 -}- x cos d; ) ,. . >. 
 
 y' = y cos 6 — X sin d;) 
 
 which are the required expressions. 
 
 To express the old co-ordinates in terms of the new co- 
 ordinates, we have 
 
 X = X* cos S — y^ sin d; 
 y = a;' sin (J -f- y' cos 
 
 If we take for (^ the angle which the new axis of Y makes 
 with the old axis of X, the new axis of Xwill make an angle 
 oi d — 90° with the old one. Hence in this case the formulae 
 of transformation will be found by writing 6 — 90° for d in 
 (12), which gives 
 
 8;\ 
 
 s.S 
 
 (13) 
 
 X = x' sin S -\- y* cos d\ ) 
 
 y — — o:^ cos S -\- y' sin 6. 
 
 (13) 
 
28 
 
 PLANE ANALYTIC OEOMETBY. 
 
 28. Thirdly. Let the new system of co-ordinates have 
 any origin and direction whatever, and let us put, as before, 
 
 «, J E the co-ordinates of the new origin referred to the 
 old system; 
 
 (J, the angle which each axis of the new system forms 
 with the corresponding axis of the old one. 
 
 Imagine through the new origin 0' an intermediate system 
 of co-ordinates parallel to the old system, and let us put x^ and 
 y, the co-ordinates of P referred to this intermediate system. 
 
 Then, by (10), 
 
 X^ ^^ X — a\ y^ = y — If, 
 
 By (11), 
 
 Whence 
 
 x' = y^ sin d -\- x^ cos d; 
 y' = y^ COS S — x^ sin d. 
 
 = {y — h) sin S -\- {x — a) cos 6-, \ 
 =z (y — h) cos 6 — {x — a) sin d; ) 
 
 (14) 
 
 which are the required expressions. 
 
 29. Transformation from rectangular to oblique co-ordi- 
 nates, the origin remaining the same. 
 
 Let OXy OYhe the rectangu- 
 lar axes, and 0X% OY' the ob- 
 lique axes; the angle XOX^ = a, 
 XOY' = /?; and let x, y be the 
 co-ordinates of any point P re- 
 ferred to the rectangular axes, 
 and x^, y' the co-ordinates of the 
 same point referred to the oblique 
 axes. Then 
 
 x= 0M= 0N-\- M'Q 
 = OM' cos XOX + PM' cos XO Y' 
 
 {since XOr=PM'Q) 
 
 = x' cos oc -\- y' cos ^, 
 
 and 
 
 y = PM = M'N-{- PQ 
 = OM' sin XOX' -f PM' sin XOY' 
 = x' sin a -\- y' sin /J; 
 
C0-0BDINATE8 AND LOCI. 20 
 
 which are the expressions of the rcctaiigiihir co-ordinates in 
 terms of the oblique ones. If we express the oblique co-or- 
 dinates in terms of the rectangular ones, we shall have 
 
 , X s\n /3 — 7/ cos /3 . , y cos a — x sin a 
 
 x' = ■ ,,. ^ ' - and 1/ = ^ r-7-5- r — . 
 
 sin(/i — a) ■' sm(P — a) 
 
 Of Loci. 
 
 30. The first fundamental principle of Analytic Geom- 
 etry, as developed in what precedes, may be expressed thus: 
 
 Having chosen a system of co-ordinates, then 
 
 To every pair of values of the co-ordinates corresponds one 
 definite point in the plane. 
 
 Let us now suppose that, instead of the co-ordinates beiug 
 given, only an equation of condition between them is given. 
 Then we may assign any value we please to one co-ordinate, 
 and find a corresj^onding value of the other. To every such 
 pair of corresponding values will correspond a definite point. 
 Since these pairs of values may be as numerous as we please, 
 we conclude: 
 
 A pair of co-ordinates subjected to a single equation of 
 condition may belong to a series of points imlimited in num- 
 ber. 
 
 If one co-ordinate varies continuously and uniformly, the 
 other will vary according to some regular law. From this 
 follows: 
 
 The poi7its whose co-ordinates satisfy an equation of co7i- 
 dition all lie on one or more lines, straight or curved. 
 
 Def A line, or system of lines, the co-ordinates of every 
 point of which satisfy an equation of condition is called the 
 locus of that equation. 
 
 3 1 . Problem. To draw the locus of a given equation. 
 Solution. 1. By means of the equation express one co- 
 ordinate, no matter which, in terms of the second. 
 
 2. Assign to this second co-ordinate a series of values, at 
 pleasure, differing not much from each other. 
 
30 
 
 PLANE ANALYTIC GEOMETRY. 
 
 3. Find each corresponding value of the other co-ordinate. 
 
 4. Lay down the point corresponding to each pair of values 
 thus found, and join all the points by a continuous line. 
 
 5. This line will be the required locus. 
 Example 1. Construct the locus of the equation 
 
 lOy = x' - X - ^. 
 
 Assigning to x values from — 10 to + 10, differing by 
 two units, we have 
 
 a; = - 10 I 
 
 8 1-6 
 6.814-3.8 
 
 + 
 
 -2 1 1+2 14-4 1+6 1+8 14- 10 
 + .21- .41- .21+ .81+2.61+5.21+ 8. 
 
 Laying down the positions of these eleven points corre- 
 sponding to these pairs of co-ordinates, we find them to be as 
 in the annexed diagram. 
 
 X 
 
 M 
 
 Example 2. Construct the locus of the equation 
 
 (y - ^y 4- (^ - 13)' = 100. 
 
 From this quadratic equation we obtain for the value of y, 
 in terms of x, 
 
 y = 5 ± VIOO - (a; - 12)'. 
 
 The following conclusions follow from this equation: 
 1. For every value we assign to x there will be two values 
 of y, the one corresponding to the positive, the other to the 
 negative value of the sum. To form the locus we must lay 
 down both of these values. 
 
CO-ORDINATES AND LOCI, 
 
 31 
 
 2. If the value of {x 
 
 case when 
 
 a; < 3 or x > 
 
 • 12)"^ exceeds 100, which will be the 
 22, the quantity under the radical 
 
 sign will be negative, and the value of y will be imaginary. 
 This shows that there is no value of ?/, and therefore no point 
 of the curve, except when x is contained between the limits 
 
 2 < .T < 22. 
 
 We now find the following sets of corresponding values of 
 X and y: 
 
 2 
 
 3 
 
 4 G 
 
 8 
 
 10 
 
 12 
 
 14 
 
 16 
 
 18 
 
 20 
 
 21 
 
 23 
 
 5.0 
 
 9.4 
 
 11.0 13.0 
 
 14.2 
 
 14.8 
 
 15.0 
 
 14.8 
 
 14.2 
 
 13.0 
 
 11.0 
 
 9.4 
 
 5.0 
 
 5.0 
 
 0.6 
 
 -1.0 -3.0 
 
 -4.2 
 
 -4.8 
 
 -5.0 
 
 -4.8 
 
 -4.2 
 
 -3.0 
 
 -1.0 
 
 0.6 
 
 5.0 
 
 Laying down these points upon a diagram, we shall find 
 them to fall as in the annexed figure. 
 
 EXERCISES. 
 
 Construct the loci of the following equations to rectangu- 
 lar co-ordinates: 
 
 1. y = Zx^ - X -10. 
 
 2. ^ = sin X. 
 
 3. y = cos X. 
 
 Note. Iu the last two exercises we should, in rigor, adopt the unit 
 radius, 57° 18', as the unit of x. But a more convenient and equally 
 good course will be to take 60° as the unit, and let it correspond to one 
 inch on the paper. Lay off a scale of sixths of an inch on the axis of X, 
 and let the successive points, one sixth of an inch apart, be 0°, 10°, 20°, 
 
82 PLANE ANALYTIC GEOMETRY. 
 
 30% 360°. At each poiut erect, as an ordinate, the corie- 
 
 spouding value of the natural sine or cosine, and draw the curve through 
 the extremities. The curve is called the curve of sines. 
 
 We need not stop at 360°, but may continue on indefinitely. The 
 curve will be a wave-line, the parts of which continually repeat them- 
 selves. 
 
 4. 
 
 y =1 + 1. 
 
 
 5. 
 
 y =^x ^ 1. 
 
 
 6. 
 
 X =\y' - 3. 
 
 
 7. 
 
 5ru = ?/' - by - 
 
 5. 
 
 8. 
 
 lOx = y' -^ - 
 
 10. 
 
 9. 
 
 f =x\ 
 
 
 10. 
 
 y = tan x. 
 
 
 11. 
 
 y = sec X. 
 
 
 Note. The object of the above exercises is to give the student a 
 clear practical idea of the relation between an equation and its locus. 
 He should perform as many of them as are necessary for this purpose. 
 It is in theory indifferent what scale of units of length is used, but in 
 practice a scale either of millimetres or tenths of an inch will be found 
 most convenient. 
 
 32. Intersections of Loci. Consider the following prob- 
 lem: 
 
 To find the point or points of intersection of two loci given 
 hy their equations. 
 
 Solution. Since the points in qnestion are common to 
 both loci, their coordinates must satisfy doth equations. 
 Hence we have to find those values of the co-ordinates which 
 satisfy both equations. This is done by solving the equations 
 algebraically, regarding the co-ordinates as unknown quanti- 
 ties. 
 
 If the equations are each of the first degree, there will 
 be but one pair of values of the co-ordinates, and therefore 
 but one point of intersection. 
 
 If the equations are one or both of the second or any 
 higher degree, there may be several roots, in which case there 
 will be one point for each pair of roots. The curves will then 
 have several points of intersection. 
 
CO-ORDINATES AND LOCI. 33 
 
 If the roots arc one or both imaginary, the loci will not 
 intersect at all. This is expressed by calling the points of 
 intersection imaginary. 
 
 Example. To find the point in which the loci whose 
 equations are 
 
 3^» + 2a;' = 164 
 and y = 2x — 3 
 
 intersect each other. 
 
 We have here a pair of simultaneous equatiouSj one of 
 which is a quadratic. Substituting in the first the value of y 
 from the second, we have the quadratic equation in x, 
 
 Qx' - 12x = 155. 
 
 The solution of this equation gives 
 
 x = l±\/^---= -\-6.1S or -4.18. 
 
 The corresponding values of y are 
 
 y = 9.3G or - 11.36. 
 
 We have in this theory a correspondence between the 
 moUlity of a point in space and the variahility of an alge- 
 braic quantity, which is at the basis of Analytic Geometry. 
 That is: 
 
 To the unlwiited variability of the co-ordinates x and y 
 corresponds the mobility of a point to all parts of a plane. 
 
 To the limited variaUlity of co-ordinates subjected to one 
 equation of condition corresponds the limited mohility of a 
 point confined to a straight or curve line, but at liberty to 
 move anywhere along that line. 
 
 To the co7istancy of co-ordinates required to satisfy two 
 equations corresponds the immohility of a point required to 
 be on two lines at once, that is, confined to the intersections 
 of two lines. 
 
 It must always be understood that liberty to occupy any 
 one of several points, as when the curves have several points 
 of intersection, is not mobility. 
 
34 PLANE ANALYTIC GEOMETRY. 
 
 EXERCISES. 
 
 Find the points of intersection of the following loci: 
 
 1. i^ + f =1 and ^-^ = 2. 
 a b ha 
 
 2. x'-{-i/ = ^ and 1+1=1. 
 
 3. 1+1=1 and x' + y'=l. 
 
 4. y^- = 4:ax and x — VSy = 2. 
 Do the following loci intersect? — 
 
 5. 3rc' - ?/' = - 4 and x'-{-y' -2x = 0. 
 
 X 2 ^ 
 iKf "^ 3 and 9^;^ + 2bi/ = 225. 
 
 Note. The special values of the coordinates found from the above 
 exercises are constants, the relation of which to the variables may be 
 explained by thinking thus : The co-ordinates are affected by a love of 
 liberty which prompts them to take all possible values so long as we, 
 their masters, do not subject them to any condition. 
 
 If we require them to satisfy an equation, they obey us, but exercise 
 their liberty by assuming all values consistent with that equation. 
 
 If we require them also to satisfy a second equation, we deprive them 
 of all liberty of variation, and chain them down to the special values 
 which satisfy both equations. 
 
 Again, if we put 
 
 P = ax -\- by -\- c, 
 
 then, so long as we require the co-ordinates to satisfy the equation 
 P= 0, P retains this zero value. But if we rub out the = 0, and leave 
 only the symbol P without any equation, the co-ordinates instantly re- 
 sume their liberty, and, by varying, make P take all values whatever. 
 
CHAPTER III. 
 
 THE STRAIGHT LINE. 
 
 Section I. Elementary Theory of the Straight 
 
 Line. 
 
 The Equation of a Straight Line. 
 
 33. Problem. To find the equation of a straight line. 
 
 In order that the equation of any locus may be found, the 
 locus must be so described that the position of each of its points 
 can be determined. Hence, to find the equation of a straight 
 line, we must suppose the data which determine the situation 
 of the line to be given. This may be done in various ways, 
 of which the following are examples: 
 
 I. A line is completely deter- y 
 mined if the point in which it 
 intersects the axis of X, and the 
 angle which it makes with that 
 axis, are given. Let us then 
 suppose given : 
 
 The abscissa OR = a oi the 
 point E in which the line inter- 
 sects the axis of X; 
 
 The angle e which the line makes with that axis. 
 
 To find the equation, let P be any point whatever on the 
 line. From P drop the perpendicular P3f upon the axis of 
 X. If we then put 
 
 X, y, the co-ordinates of P, 
 we shall have 
 MP = y = EM tan s = {OM - OE) tan e = (x - a) tan 8. 
 
36 
 
 PLANE ANALYTIC GEOMETRY. 
 
 Hence, putting m ee tan £, we have 
 
 y — 7n{x — a). (l) 
 
 Because P may be any point Avhatever on the line, tins equa- 
 tion must subsist between the co-ordinates of every point of 
 the line; it is therefore the equation of the line. 
 
 Def. The slope of a line is the tangent of the angle which 
 it forms with the axis of abscissas. 
 
 II. Let the slope in of the line, and the ordinate h of the 
 point in which the line cuts the y 
 axis of I', be given. 
 
 Using the same notation as be- 
 fore, we readily find 
 
 MP = X tan e -{- b, 
 or y = 7nx -\- b\ (2) 
 
 which last is the required equation. 
 
 III. Let the points A and B in which the line intersects 
 the axes of co-ordinates be given. Let us then put 
 
 a = the abscissa OA of the 
 l^oint A in which the line inter- 
 sects the axis of X; 
 
 b = the ordinate of the point 
 B in which it intersects the axis 
 of F. 
 
 Then, if P be any point on 
 the line, the similar triangles 
 BOA and P3IA give the pro- 
 portion 
 
 or 
 
 b : a = 
 
 : FM : MA = y 
 
 iVe hence derive 
 
 
 
 ay = b{a- x); 
 
 is, 
 
 bx -\- ay -— ab, 
 
 
 a^ b ^' 
 
 (3) 
 
 Def. The lengths OA and OB from the origin to the 
 points in which the line cuts the co-ordinate axes are called 
 the intercepts of the line upon the respective axes. 
 
THE STBAIQHT LINE. 37 
 
 EXERCISES. 
 
 1. Write the equations of lines passing through the origin 
 and making the angles of 45°, 30°, 120°, 135°, 150° and e, 
 respectively, with the axis of X, 
 
 2. If the intercept of a line on the axis of X is a, and if 
 € is the angle which it makes with that axis, express its inter- 
 cept on the axis of Y, 
 
 3. Write the equation of the line whose intercept on the 
 axis of J^ = 5 and which makes an angle of 30° with the 
 axis of X. 
 
 4. Form the equation of the line whose intercept on the 
 axis of X is a and which makes an angle of 45° with that 
 axis. 
 
 5. Show geometrically that tlie inverse square of the per- 
 pendicular from the origin upon a line is equal to the sum of 
 the inverse squares of its intercepts on the axes. 
 
 Note. The inverse square of a is 1 -4- a'. 
 
 6. Express the tangents of the angles which a line makes 
 with the co-ordinate axes in terms of its intercepts upon tliose 
 axes, and explain the algebraic sign of the tangent. 
 
 7. Two lines have the common intercept a upon the axis 
 
 of X; the difference of their slopes is unity; and the sum of 
 
 their intercepts upon the axis of Y is c. Find the separate 
 
 intercepts upon ?/, and show that the equations of the two lines 
 
 are 
 
 X 2?/ X 2?/ ^ 
 
 - H = 1 and - -\ ~— = 1. 
 
 a c — a a c -\- a 
 
 8. What is the relation of the two lines, 
 
 y = mx -{- b 
 and y = mx -{- b -\- c? 
 
 9. What are the relations of the series of lines, 
 
 y = mx, 
 y = mx -\- by 
 y z= mx -\- 2b, 
 y = mx -f 3b, 
 etc. etc. ? 
 
38 PLANE ANALYTIC GEOMETRY. 
 
 10. What is the relation of the two lines, 
 1/ = b -\- mx 
 and y -z^ h — mx"^ 
 
 Especially show where they intersect, the relation of the 
 angles they form with tlie axis, and the angle they form with 
 each other in terms of ^ = arc tan m. 
 
 34. The eqnations (1), (2) and (3) are examples of the 
 nnmerous forms which the equation of a right line may 
 assume. "We have now to generalize these forms. 
 
 Def. An equation of the first degree between two 
 variables, x and y, means any equation which can be reduced 
 to the form 
 
 Ax-\-By^G=0, (4) 
 
 A, B and C being any constant quantities whatever. 
 
 Theorem. Every eqication of the first degree between rect- 
 angular co-ordinates represents a straight line. 
 
 Proof. The equation (4) may be reduced to the form 
 
 .= -^(. + 5). (5) 
 
 Since the tangent of a varying angle takes all values, we 
 
 can always find an angle, = s, whose tangent shall b& 
 
 B' 
 
 Q 
 
 On the axis of X measure a distance ^ = a. 
 
 A 
 
 Then, by (1), the locus of (5) will be the line which inter- 
 sects the axis of J^ at the point x = a and makes an angle e 
 with the axis of X. Since such a line is always possible, the 
 theorem is proved. 
 
 Scholium. The result of the above theorem may be ex- 
 pressed as follows: 
 
 The locus of the equation 
 
 Ax-[-By-{- C=0 
 is that straight line which intersects the axis of X at the 
 
 C 
 
 distance — -r from the origin and makes with that axis an 
 
 a7igle whose tangent is — ^. 
 
THE STRAIGHT LINE. 39 
 
 35. Reductioji of the General Equation. Any pair of 
 values of x and y which satisfy the equation 
 
 ^a; -f- % + C = 
 must also make 
 
 m{Ax + % + C) = 0; 
 
 that is, 
 
 {mA ) X + {7nB)y -{- wC = 0. 
 
 Hence, since m may be any quantity whatever. 
 
 If we multiiily or divide all the coefficients, A, B and C, 
 xohich enter into the general equation, hy the same factor or 
 divisor, the line represented ly the equation will not he altered. 
 Example. The equations 
 
 y — 2.1- + 1 = 0, 
 2y — 4:3; + 2 = 0, 
 5?/ - 10a; 4- 5 = 0, 
 
 all represent the same line, because they all give the same 
 value of y in terms of x, namely, 
 
 y = 2x - 1. 
 
 The same result may be expressed in the form : 
 
 The line represented ly the equation (4) depends only on 
 
 the mutual ratios of the coefficients A, B and C, and not 
 
 upon their absolute values.'^ 
 
 36. From this it follows that special forms of the general 
 equation may be obtained by multiplying or dividing it by any 
 quantity. 
 
 I. First Form. By dividing by B we obtain 
 
 5^ + y + S = o, 
 
 or AC .^. 
 
 * This introduction of more quantities than are really necessary for 
 the expression of a result is quite frequent in Mechanics and Geometry. 
 It has the advantage of enabling us to assign such values to the super- 
 fluous quantities as will reduce the expression to the most convenient 
 form. 
 
40 PLANE ANALYTIC GEOMETRY. 
 
 which becomes identical with tlie form (2) by putting 
 
 
 ™h-|; iH-J (7) 
 
 11. Second Form, By dividing by C the general equation 
 becomes 
 
 
 ^x + -^y + l=:d, (8) 
 
 or 
 
 A B 
 
 which becomes identical witli (3) by putting 
 
 
 , 
 ''--A' ^^-B- 
 
 III. Third or Normal Form. Let us divide by VA^-\- B\ 
 The equation will then become 
 
 ABC 
 
 ^ + -:7^^=f=%^y + ^7^F^^=0. (10) 
 
 VA"" + B' VA' -\- B' VA'^ b 
 
 If we now determine- an angle a by the equation 
 
 B 
 
 sin a 
 
 VA' + B' 
 we shall have 
 
 cos a = Vl — silica = — . (H) 
 
 Va' + b' ^ ^ 
 
 Let us also put, for brevity, 
 
 O 
 
 P = 
 
 VA' + B' 
 
 The general equation of the line will then become 
 
 a: cos a + y sin a — jt? = 0, (12) 
 
 which is called the Normal form of the equation of a 
 straight line. 
 
THE STRAIGHT LINE. 41 
 
 EXERCISES. 
 
 Express each of the following equations in the forms (2), 
 (3) and (10): 
 
 1. 3a; + 4?/ + 15 = 0. 2. 4a: -f- 3?/ - 15 = 0. 
 3. 12x - 5?/ - 13 = 0. 4. X -2y -{- 6 = 0. 
 5. X -{- y -\- c = 0. 6. X — y — c = 0. 
 
 3 7 . Relation of the General Equation to its Special Forms. 
 The forms (1), (2) and (3) are examples of numerous special 
 forms under which the equation of a straight line may be 
 written. The general form is not to be regarded as a distinct 
 form, but as a form which may be made to express all others 
 by assigning proper values to the constants A, B and 0. 
 For example: 
 
 The form (1) is equivalent to 
 
 y — mx + ma = 0, 
 which is what the general form becomes when we put 
 
 A = — m, 
 B=l, 
 = am. 
 In the same way, to reduce the general form to (2), we 
 
 have only to put 
 
 A = — 7n, 
 
 B=l, 
 
 C=-h. 
 
 To reduce it to (3) we put 
 
 . (7=1. 
 Again, the normal form is one expressed by the general 
 form when we suppose 
 
 A = cos a, 
 B = sin a, 
 
42 
 
 PLANE ANALYTIC GEOMETRY. 
 
 "We may also say that the normal form is one in which 
 A' ^ B' = 1. 
 
 38. Since all forms of the equation of a straight line are 
 special cases of the general form, we conclude: 
 
 If we demonstrate any theorem hy means of the general 
 form of the equatmi of a straight line, that demonstration tuill 
 include all the special forms. 
 
 39. Def. The constants A, B and C which enter into 
 the equation of a straight line are called its parameters. 
 
 The parameters determine the situation of a line as co- 
 ordinates do the position of a point. 
 
 Only two parameters are really necessary to determine the 
 line, but there is often a convenience in using three, as in the 
 general form. 
 
 A line is completely determined when its parameters are 
 given. Instead of saying, 
 
 *' The line whose equation \q Ax -\- By -\- C — 0," 
 we may say, 
 
 "The line (^1, B, (7)." 
 
 40. Special Cases of Straight Lines. 
 
 I. If, in the general equation of the straight line, 
 
 Ax -\- By -^ C = 0, 
 
 the coeflBcients A and B are 
 of opposite signs, x must in- 
 crease with y, the line makes 
 an acute angle with the axis of 
 X, and its positive direction isQ' 
 in the first or third quadrant. 
 QR is such a line. 
 
 II. If A and B are of the 
 same sign, one co-ordinate diminishes as the other increases, 
 the line makes an obtuse angle with the axis of X, and its 
 positive direction is in the second or fourth quadrant. 
 
 PS is such a line. 
 
THE STRAIGHT LINE. 43 
 
 III. If A vanishes, the equation may be reduced to 
 
 y = 7> ~ ^ constant, 
 
 while X may have any value whatever. 
 
 The line is then parallel to the axis of X and at the dis- 
 tance — -^ from it. 
 
 IV. In the same way, the equation of a line parallel to 
 
 the axis of Y is 
 
 a; = a constant, 
 
 the constant being the distance of the line from the axis of Y. 
 
 V. If this constant itself vanishes, the line will coincide 
 with the axis of Y. Hence the equation of the axis of y is 
 
 X = 0. 
 
 VI. In the same way, the equation of the axis of x is 
 
 y = 0. 
 
 EXERCISES. 
 
 1. At what point does the line 
 
 ax -\- c = 
 cut the axis of X? 
 
 2. Write the equation of a line perpendicular to the axis 
 of Xand cutting off an intercept, d, from that axis. 
 
 3. What are the relations of the four lines, 
 
 X = a-, X = — ct'y 
 
 y = h y = -h 
 
 and what figure do they form? 
 
 41. Special Problems co?mectecl with the General Equa- 
 tion of a Straight Line. 
 
 I. To find the iiitercejjts of the general straight line iipon 
 the co-ordinate axes. 
 
 By definition, the intercept upon the axis of X is the value 
 of X when y = 0. Putting y = \n the general equation, it 
 becomes 
 
 Ax +(7=0. 
 
44 
 
 PLANE ANALYTIC OEOMETRT. 
 
 a = 
 
 Hence, if we put a for the intercept upon the axis of X, 
 we have 
 
 C 
 A' 
 
 In the same way, we find for the intercept on Y, which we 
 call b, 
 
 II. To find the angle which a line makes with the axis of X. 
 We have already shown (§ 34) that 
 
 A 
 B' 
 
 We can now find the sine and cosine of e by trigonometric 
 formulae, as follows: 
 
 tan e A 
 
 tan e 
 
 sm £ = 
 
 cos € = 
 
 Vl + tan^f 
 1 
 
 VA' + B 
 B 
 
 (13) 
 
 VI + tanV VA"" + B^ 
 
 III. To ex2)ress the i^erpendicular distance of a 'point from 
 a given line. 
 
 Let X* and y' be the co-ordinates of the point, and 
 Ax-^ By -^ C -^ 
 the equation of the line. 
 
 Since the position of the point is completely determined 
 by its co-ordinates, and the line 
 by its parameters, A, B, C, the 
 required distance admits of being 
 expressed in terms of x', y', A, 
 B and C. 
 
 Let P be the point, LN the 
 line, and PQ the perpendicular 
 from the point on the line; and 
 let the ordinate PM of the point 
 intersect the line in 11. We shall . 
 then have ^'^ 
 
 PQ ^ PE cos e. (a) 
 
 Y 
 
 
 P 
 
 
 
 
 K\ 
 
 
 
 / 
 
 5>r 
 
 "^ 
 
 
 
 /Q 
 
 M. 
 
 
THE STRAIGHT LINE. 45 
 
 Now R is a point on tlie line whose abscissa is the same as 
 that of P, namely, x'] and if we put RM = y^ = the ordi- 
 nate of R, we must have, since R is on the line, 
 
 Ax^ -{. Bt/, -^ C = 0, 
 
 , . , . Ax' -i- C 
 which gives y^ = h • 
 
 Then PR = PM - RM = y' - y^ 
 
 _B£_ _ Ax' -{. By' -^r O 
 ~ B ^' ~ B 
 
 Substituting in {a) this value of PR and the value of cos e 
 
 from (13), we have 
 
 ^ VA' -{- B' ^ ^ 
 
 Since the co-ordinates of the origin are x' = and y' = 0, 
 we have 
 
 OQ' = , ^ -- (15) 
 
 VA' -{- B' ^ ^ 
 
 which gives the perpendicular from the origin on the line. 
 
 EXERCISES. 
 
 Find, for each of the lines represented by the following 
 equations, — 
 
 The angle which it makes with the axis of X; 
 
 Its intercepts upon the axes; 
 
 Its distance from the point (4, 3); 
 
 Its least distance from the origin; 
 
 The length of that portion, intercepted between the axes. 
 
 1. 3a; + 4?/ + 10 = 0. 2. Sx + 4?/ - 10 = 0. 
 
 3. 6x — 12y 4- 26 = 0. 4. a; + ?/ = 0. 
 
 5. 4:X — 3y — 5 = 0. 6. x — y = 0. 
 
 X 11 
 
 7. X COB, a -\- ij Bin OL — p = 0. 8. ^ — h t- = 1. 
 
 9. Find the length of the perpendicular from the point 
 
 X II 
 
 {a, I) on the line — -j- ^ = 1, and show that it is equal to 
 the negative distance of the line from the origin. 
 
46 
 
 PLANE ANALYTIC GEOMETRY. 
 
 10. Find tlie points on tlie axis of X which are at a per- 
 pendicular distance a from the line — j- •- — 1 = 0. 
 
 42. Direct Derivatioji of the Normal Form. This form 
 may be derived as follows: y 
 From the origin drop the per- 
 pendicular OM upon the line 
 whose equation is required. 
 
 Let P be any point of the 
 line, and 
 
 X = ON, the abscissa of P; 
 
 y = NP, its ordinate; 
 
 a = angle NOMoi the per- 
 pendicular with the axis of X\ 
 
 P= OM. 
 
 From N draw NQ parallel to the line, and PR parallel to 
 OM. Then 
 
 OQ = OiV^cos a = X cos a; 
 QM = NP sin a = y sin a; 
 
 Hence 
 
 OQ -\- QM = p — X cos a -\- y sin a. 
 
 X cos a -\- y sin a — p — 0, 
 
 which is the normal form of the equation. 
 
 We hence conclude: 
 
 In the normal form the parameters p and a arc respectively 
 the perpendicular from the origin upon the line, and the angle 
 lohicli this p)erpendicidar makes with the axis of X. 
 
 43. Distances from a Line in the Normal Form. In this 
 form A"^ -\- B^ = 1. Hence the distance of the point whose 
 co-ordinates are x' and y' from the line is 
 
 x' cos a -\- y' sin a — p 
 
 the same function which, equated to zero, represents the line. 
 Hence the theorem: 
 
 If, in the expression x cos a -\- y sin a — p, we suhstitiite 
 for X and y the co-ordinates of any point whatever, the expres- 
 
THE STRAIGHT LINE. 4,1 
 
 sion luill represent the distance of that point from the line whose 
 equation is x cos oc -\- y sin a — p =: 0. 
 
 By supposing x' and y' zero, we find the distance of the 
 origin from the line to be — p. Since p itself has been taken 
 as essentially positive, we conclude: 
 
 The expression for the distance of a point from the line in 
 the normal form is negative luhen the point is 07i the same side 
 as the origin, and positive on the opposite side. 
 
 This agrees with the convention that the direction /ro??i 
 the origin to the line shall be positive. 
 
 EXERCISES. 
 
 1. What is the relation of the two lines 
 
 X cos 30° + y sin 30° - ^ = 
 and X cos 210° + y sin 310° — p = 0? 
 
 2. Draw approximately by the eye and hand the lines 
 represented by the following equations: 
 
 X cos 30° + y sin 30° - 5 = 0. 
 X cos 60° + y sin 36° - 5 = 0. 
 X cos 120° + y sin 120° -5 = 0. 
 X cos 210° + y sin 240° -5 = 0. 
 
 Lines Determined by Given Conditions. 
 
 When a line is required to fulfil certain conditions, those 
 conditions must be expressed algebraically by equations of con- 
 dition involving the parameters of the line. The values of the 
 parameters are to be eliminated from the equation of the line 
 by means of these equations of condition. 
 
 Since two conditions determine a line, it will be convenient 
 to employ a general form of the equation of the line in which 
 only two parameters appear. Such a form is 
 
 y = mx + h. (a) 
 
 44. To find the equation of a line ivhich shall pass through 
 a given p)oint and malce a given angle with the axis of X. 
 
 Let {x', y') = the given point, and 
 
 f = the given angle. 
 
48 
 
 PLANE ANALYTIC GEOMETRY. 
 
 One equation of condition is then 
 
 m = tan e, 
 
 which determines the parameter ??^. This gives, for the equa- 
 tion of the line, 
 
 y = X tan £ -{- h. (b) 
 
 The condition that the line shall pass through the point 
 {x', y') is 
 
 ?/' z= mx' -\- h. 
 
 To eliminate hy we subtract this equation from {b) after 
 substituting the value of 7n. This gives 
 
 y — y' — tan ^ {x — a:'), 
 
 which is the required equation of the line passing through the 
 point {x' , y') and making an angle e with the axis of X, If 
 we write m for tan e, it becomes 
 
 or 
 
 y - y 
 
 mx — y 
 
 - m(x — x'), 
 mx' + ^/ = 0, 
 
 (c) 
 
 which, compared with the general form 
 Ax -\- By -\- C = 0, 
 
 ^= - 1; 
 
 gives 
 
 m: 
 
 C = - 77ix' 4- y\ 
 
 45. To find the equation of a line passing through tiuo 
 given 2Joints. 
 
 Let {x^y y^ and (x^, yj be the two given points. 
 To determine the param- 
 
 eters m and b, we have the 
 
 conditions 
 
 
 
 y. = 
 y.= 
 
 mx^ + b, I 
 mx^ -\-b,) 
 
 (d) 
 
 which give, 
 
 by subtraction 
 
 > 
 
 y.-y^ 
 
 = (^a - ^d'tn ; 
 
 
 •whence m 
 
 _y^- y. 
 
 X^ — X, 
 
 (e) 
 
THE STRAIOHT LINE. 49 
 
 Subtracting the first equation of {d) from (a), we have 
 
 y -y. = H^ - ^,), 
 and substituting the vahie of m gives 
 
 2/ - y. = ■F^f"(^ - *.)' (16) 
 
 *^2 — ^1 
 
 which is the required equation in which the parameters m and 
 h are replaced by the co-ordinates of the given points. 
 
 To reduce the equation to the general form, we have, by 
 clearing of denominators, 
 
 ■A = y.- .Vi; ) 
 
 B=x^- X,', \ (17) 
 
 G = yX^2 - ^i) - ^i(^3- y.) = ^^y- ^.y^- ' 
 
 Remark. Most of the special forms of the equation already given 
 are cases in which the line is determined by given conditions. For 
 example: 
 
 In the form (1) (§ 33) the given quantities are the slope and the inter- 
 cept on the axis of X 
 
 In the form (2) they are the slope and the intercept on the axis of Y, 
 
 In the form (3) they are the two intercepts. 
 
 In the Normal form they are the length of the perpendicular from 
 the origin upon the line, and the inclination of the perpendicular to 
 the axis of X 
 
 EXERCISES. 
 
 1. Write the equation of a line passing through the point 
 (—1, 2) and making an angle of 135° w^ith the axis of X. 
 
 2. Write the equation of a line passing through the point 
 (4, —1) and making an angle of 30° with the axis of X; find 
 the intercepts which it cuts off from the axes, and the ratios 
 of these intercepts to the length of the line included between 
 the axes. 
 
 3. Find the equation of the line passing through the points 
 (2, 4) and (3, —2), and find its intercepts on the axes, the angle 
 which it makes with the axis of X, and its distance from the 
 origin. 
 
 4. Find the equation of the line making an angle of 150° 
 with the axis of X and passing at a perpendicular distance 5 
 from the origin. 
 
50 PLANE ANALYTIC GEOMETRY. 
 
 5. Find the equiiuxon of die line passing through the point 
 (1, 5) and intercepting a length 3 on the axis of Y. 
 
 G. Find the equation of the line passing through the point 
 (5,-1) and intercepting a length — 3 on the axis of Y. 
 
 7. Write the equations of lines passing through the three 
 following pairs of points: 
 
 I. The points {a, h) and {a, — b), 
 II. The points (— a, h) and {a, h). 
 III. The points («, h) and (— a, — h). 
 
 8. What is the distance from the point (1, 5) to the line 
 joining the points (— 3, 3) and (1, 6)? Ans. — . 
 
 
 
 9. If the Yertices of a triangle are at the points (1, 3), 
 (3, — 5) and (— 1, — 3), write the equations of the three 
 sides in the general form, and find the distance at which each 
 side passes from the origin. 
 
 10. Write the equations of the three medial lines of this 
 last triangle. 
 
 Note. A medial hue of a triangle is the line from either vertex to 
 the middle of the opposite side. 
 
 11. Given the co-ordinates of the vertices of a triangle, 
 find the equations of the lines which join the middle points 
 of any two sides, and show that these joining lines are parallel 
 to the sides of the triangle. 
 
 12. Find the equations of the three sides of the triangle 
 whose vertices are at the points {a, h), (a', I') and a", h"). 
 Then find the product of the length of each side into its dis- 
 tance from the opposite vertex, and show that each of these 
 products is equal to the double area of the triangle. 
 
 First write the general equation of each side, using the form (17). 
 Then note the relation between each value of ^/A^ + B'^ and the corre- 
 sponding side of the triangle. Then form the products and note § 22. 
 
 13. Show analytically that if a series of parallel lines are 
 equidistant, they contain between them equal segments of the 
 axes of co-ordinates. Note that the values of ^) for such lines 
 are in arithmetical progression. 
 
THE STRAIGHT LINE. 61 
 
 Relation of Two Lines. 
 
 46. Problem. To express the angle between two lines in 
 terms of the parameters of the lines. 
 
 Let the lines be 
 
 Ax -\- By -^ C = I . . 
 
 and A'x + B'y-\- G'= 0. i ^""^ 
 
 The angle between them will be the difference of the angles 
 which they make with the axis of X] that is, using the pre- 
 vious notation, it will be £ — e\ 
 
 The expression for the tangent oi s — e' will be the 
 simplest. We have, by trigonometry, 
 
 . . ,, tan s — tan f' ,._. 
 
 Substituting the values of tan e and tan a' found from 
 (13), this equation becomes, by reduction, 
 
 tan (. - O == 5^Tqr^-,. (19) 
 
 Or, if we put, as before, 
 
 m E tan e, m' = tan b\ 
 the expression will be 
 
 tan (* - £') = '^^^^-r (30) 
 
 1 + mm' ^ ' 
 
 Either of the forms (18), (19) and (20) is a solution of the 
 problem. 
 
 47. The following are special cases of the preceding 
 general problem: 
 
 I. To find the condition that two lines shall ie parallel. 
 This condition requires that we have 
 
 f - £' = 0° or 180°; 
 that is, 
 
 tan {e — a') — 0. 
 
52 PLANE ANALYTIC GEOMETRY. 
 
 Hence, from (20), the required condition is 
 A'B - AB' = 0, 
 
 A^ A [ (31) 
 
 or jy, = -jj. 
 
 II. To find the condition that two lines shall he perpen- 
 dicular to each other. 
 
 The lines will be perpendicular when 
 
 £ - e' = ± 90°. 
 Then 
 
 tan (£ — 6') = 00 . 
 
 In order that the second members of either of the equa- 
 tions (19) or (20) may become infinite, its denominator must 
 be zero. Hence we must have 
 
 AA^ + BB' = 0, ) .^^. 
 
 or 1 + mm' = 0, ) ^ ' 
 
 or tan e tan e' = — 1, 
 
 * which are three equivalent forms. 
 
 EXERCISES. 
 
 Write the equations of the lines passing through the origin 
 and perpendicular to each of the following lines: 
 
 1, ax -{■ by -{- c = 0. Ans. hx — ay — 0. 
 
 2. y = mx -\- h. 3. a{x -\- y) — h(x — y) = 0. 
 4. X + ny = c. 5. (x - x^) = 7)i{y - y^). 
 
 6. Write the equation of the line passing through the point 
 (a, h) and perpendicular to the line 
 
 Ax + % + C = 0. 
 
 7. Write the equation of the line through the point {a, b) 
 parallel to the line 
 
 Ax-{- By -{• C = 0. 
 
 8. Express the tangent, sine and cosine of the angle 
 between the lines 
 
 ax -\- by -\- c = 0; 
 ax — by '\- c = 0, 
 
THE STRAIGUT LINE. 53 
 
 9. Write the equations of two lines passing through the 
 origin, and each making an angle of 45° with the line 
 
 ax -\- hy -\- c =^ 0, 
 
 Ans. (a -\- b) X — {a — I)) y — 0, 
 and {a — b) X -{- {a -\- b) y = 0. 
 
 10. Compute the interior angles of the triangle the equa- 
 tions of whose sides are 
 
 a; - 3y + 7 = 0; 
 X ~\- y — d = 0; 
 X -{-dy — 0. 
 
 11. If the co-ordinates of the three vertices of a triangle 
 are (2, 5), (2, — 3), (4, — 1), it is required to find the equa- 
 tions of the three perj^endiculars from the vertices upon the 
 opposite sides. 
 
 12. Find the equations of the perpendicular bisectors of 
 the sides of the same triangle. 
 
 13. Show that the lines joining the middle points of the 
 consecutive sides of a quadrilateral form a parallelogram. 
 
 To do this assume symbols for the co-ordinates of the four vertices; 
 then express the middle points of the sides by § 23, and then the equa- 
 tions of the joining lines by § 45, and show that opposite lines are 
 parallel. 
 
 14. Find the condition that the lines 
 
 X cos a -\- y mi a — p = and x sin /3 ^ y cos /> — ^;' 
 may be parallel. 
 
 15. If two lines intersect each other at right angles, and if 
 a and b be the intercepts of the one line, and a' and b' of the 
 other, it is required to show: 
 
 (a) That of the four quantities, a, b, «', &', either three 
 will be positive and one negative, or three negative and one 
 positive. 
 
 (/?) That these quantities satisfy the condition 
 aa' + bb' = 0. 
 
 16. "What is the rectangular equation of the line whose 
 polar equation is 
 
 n 
 
 - = 4 COS + 3 sin 6? 
 
54 PLANE ANALYTIC GEOMETRY. 
 
 17. Find the area of the triangle formed by the straight 
 
 lines 
 
 y = X tan 75°, y = x, y = x tan 30° + 2. 
 
 18. Kcduce 3r cos 6 — 2r sin ^ = 7 to the form 
 
 r cos (6 — a) = ]?, 
 
 and find the values of a and ^j. 
 
 19. Show that if F - a' = 1, the lines 
 
 X -\- (a-\-b)y+c = and {a + b)x -f- {a' - h')y -\- d = 
 
 are perpendicular to each other. 
 
 20. Show that when the axes are oblique, the ratio x : y 
 of the two co-ordinates of a point is equal to the ratio 
 
 Dist. from axis of Y : Dist. from axis of X. 
 
 21. Show that the lines x -\- y = a and x — y — a are 
 at right angles, whatever be the axes. 
 
 22. Show that the locus of a point equidistant from two 
 straight lines is the bisector of the angle they form. 
 
 48. To find the point of intersection of two lines given 
 hy their equations. 
 
 As already shown, the co-ordinates of the point of inter- 
 section are those values of x and y which satisfy loth equations, 
 (§ 32). If the given equations are 
 
 Ax -f % -f- (7 = 0, 
 A'x-\-B'y-^C' = Q, 
 "we find, for the values of the co-ordinates, 
 _ BC' - B'O 
 ^- AB' - A'B' 
 _ A'C - AC 
 ^ ~ AB' - A'B' 
 which are the required co-ordinates of the point of intersec- 
 tion. 
 
 Remark. The preceding result affords another way of 
 deducing the condition of parallelism by the condition that two 
 lines are parallel when their point of intersection recedes to 
 infinity. Let the student find this as an exercise. 
 
THE STBAIOHT LINE. 55 
 
 49, To find the condition that three straight lines shall 
 intersect in a poi7it. 
 
 The required condition must be expressed in the form of 
 
 an equation of condition between the nine parameters of the 
 
 three lines. Let the equations of the lines be 
 
 ax -{-by -\- c =0; 
 
 a'x + h'y + c' =0; 
 
 a''x + h''y + c" = 0. 
 
 If the three lines intersect in a point, there must be one 
 pair of values of x and y which satisfy all three equations. 
 By the last section we have, for the co-ordinate y of the point 
 of intersection of the first two lines, 
 
 _ a'c — ac' 
 y "~ aV - a'V 
 and of the last two, 
 
 _ a"c' - a 'c" 
 y - a'h" - a"V 
 If the three lines intersect in a point, these values of y 
 must be equal. Equating them and reducing, we find 
 
 c{a'l" - a"V) + c\a"l - ah") + c'\ah' — a'h) = 0, 
 which is the required equation of condition. 
 
 EXERCISES. 
 
 1. Given the three lines 
 
 x-\-2y^4: = 0, 
 
 2x- y -H = 0, 
 
 dx-^ y-\-c = 0, 
 
 it is required to determine the constant c so that the lines 
 
 vshall intersect in a point, and to find the point of intersection. 
 
 Ans. c = ~ 3. 
 
 Point = (2, -3). 
 
 2. Express the condition that the three lines 
 
 y = mx + c, 
 y z= m'x + c', 
 y = m"x -j- c", 
 shall intersect in a point. 
 
66 PLANE ANALYTIC GEOMETRY. 
 
 3. Find the point of intersection of the two lines 
 
 y = mx -\- c, 
 y = m'x — c. 
 
 4. Prove algebraically that if two lines are each parallel to 
 a third, they arc parallel to each other. 
 
 Note. We do this by showing that from the equations 
 
 ah' - a'b = 0, 
 ab" - a"b = 0, 
 follows 
 
 a'b" - a"h' = 0. 
 
 5. If the equations of the four sides of a parallelogram 
 are 
 
 y = mx + Cy 
 y = m'x — c, 
 y = mx -\- c\ 
 y = m'x — c', 
 it is required to find the co-ordinates of its four vertices and 
 the equations of its diagonals. 
 
 Ans.) in part. Equations of diagonals: 
 
 m 4- m' 
 
 m — m 
 6. What relation must exist among a, a', m and m' that 
 the lines 
 
 y = mx -\- a, 
 y = m'x — a', 
 
 may intersect on the axis of X? Ans. a' in + am' = 0. 
 
 50. Transformation to New Axes of Co-ordinates. 
 
 By the formuloB of § 25, the equation of a line referred to 
 one system of co-ordinates may be changed to another system 
 by an algebraic substitution. 
 
 To make the change it is necessary to express the co- 
 ordinates of the original system in terms of those of the new 
 system, and to substitute the expressions thus found in the 
 equation of the locus. 
 
THE STRAIGHT LINE. 
 
 57 
 
 
 
 u 
 
 M 
 
 Example I. Let 
 
 Ax-\-By-\- C=0 {a) 
 
 be the equation of a line referred to 
 the system {X,Y), 
 
 Let it be required to refer the line 
 to a system (X', F') parallel to the 
 first and having the origin 0' at the 
 point (a, h). 
 
 By § 26, the expressions for the original co-ordinates in 
 terms of the new ones will be 
 
 X — X* -f- a\ 
 y = //' + b. 
 
 Substituting these values in the equation (a), we find for 
 the equation of the line, in terms of the new co-ordinates, 
 Ax' + By' -^Aa -{- Bb -\- C = 0. 
 
 The coefficients A and B of the co-ordinates remain un- 
 changed, showing that the line makes the same angle with the 
 new axes as with the old ones. 
 
 Example II. Let the new system of co-ordinates have 
 the same origin, but a different direction. 
 
 The equations of transformation are then (3) of § 27. 
 Substituting the values of x and y there given in the equation 
 (a) of the preceding example, we have 
 
 (A cos d + ^ sin S)x' -{- {B cos S - A sin S)y' + C = 0. 
 
 The sum of the squares of the coefficients of a;' and y' 
 reduces to A^ -\- B^, as it should. 
 
 EXERCISES. 
 
 1. What will be the equation of the line 
 
 y = 2x -^ 5 
 when referred to new axes, parallel to the original ones, 
 having their origin at the point (2, 3)? 
 
 2. What change must be made in the direction of the axis 
 of JTthat the line whose equation isx = y may be represented 
 by the equation x' = 2y' ? 
 
68 
 
 PLANE ANALYTIC GEOMETRY. 
 
 Section II. Use of the Abbreviated Notation.* 
 
 51, Functions of the Co-ordinates. We call to mind that, 
 corresponding to any point we choose to take in the plane, 
 there will be a definite value of each of the co-ordinates x 
 and y. Hence if we take any function of x and y, such, for 
 example, as 
 
 P = ^ + 2^ + 1, 
 this function P will haye a definite value for each point of 
 the plane, which value is formed by substituting in P the 
 values of the co-ordinates for that point. We may then im- 
 agine that on each point is written the value of P correspond- 
 ing to that point. 
 
 Example. 
 
 3 4 5 6 7 8 
 
 3 4 5 6 
 
 12 3 4 
 
 -3 -2 
 
 -1 
 
 
 
 -5 -4 
 
 The above scheme shows the values of the preceding func- 
 tion P = a; -f 2?/ + 1 for a few equidistant points, assuming 
 the common distance between the consecutive numbers on 
 each line to be the unit of length. 
 
 53. Isorro2)ic Lines. We may imagine lines drawn 
 through all points for which P has the same value, and may 
 call these lines isorroinc; that is, lines of equal value. We 
 now have the theorem: 
 
 * This section can be omitted without the student being thereby pre- 
 vented from going on with subsequent chapters. But, owing to the 
 elegance of the abbreviated notation, the subject, which is not at all 
 abstruse, is recommended to all having mathematical taste. 
 
THE STRAIGHT LINE. 
 
 59 
 
 If the function P is of the first degree in x and y, the 
 isorroinc lines will form a system of j^rallel straight lines. 
 Proof. Let 
 
 P ^ax -\- by -{• c; 
 
 and let us inquire for what points P has the constant value 
 k. These points will be those whose co-ordinates satisfy the 
 condition 
 
 P - k = 0, 
 or 
 
 ax -{- by -\- c — k = 0. {a) 
 
 This equation, being of the first degree, is the equation of 
 a straight line, whose angle with the axis of X is given by 
 the equation 
 
 tan € = 7. 
 
 
 
 Since a and b retain the same values, whatever values we 
 assign to h, e has the same value for each line of the system, 
 and all the lines are parallel. 
 
 53. Distance betiveen Two Lilies of the System. To each 
 value of h in the equation {a) will correspond a certain line. 
 We now have the problem: 
 
 To find the distance betiueen the tioo lines for which P has 
 the resjjective values ^, and h^. 
 
 ir\ 
 
 Let OJf and OJVbe the respective intercepts of the lines 
 on the axis of X. We shall then have 
 
 Distance MQ = MN ^m s. 
 
60 PLANE ANALYTIC OEOMETRT. 
 
 Patting ?/ = in the two equations 
 
 ay -]-hx^c-k^ = \ 
 and ay-\-hx-{-c — lc^ = Qf,) 
 
 we have, for the intercepts, 
 
 OM ^'^^ 
 
 MN= ON- 0M = ^^-, 
 and MQ = {K - K) %^ = - ^A. . (§36) 
 
 Hence, the distance apart of two isorropic lines is propor- 
 tional to the difference between the vahies of P. 
 
 54. Distance of a Point from a Line. Let us now re- 
 turn to the general expression 
 
 P = ax -{- by -\- c, (a) 
 
 and let us study its relation to the line 
 
 ax + by -\- c = 0;) ... 
 
 that is, to the line P = 0. ) ^ ^ 
 
 In (a) we may suppose x and y to have any values what- 
 ever. But in (b) X and y are restricted to those values which 
 correspond to the different points of the line («, b, c). 
 
 Now from what has just been shown it follows that the 
 points for which, in («), P has the special value h all lie on 
 a straight line parallel to the line P = 0, and distant from it 
 by the quantity 
 
 k 
 
 Va' + b' 
 
 Hence, if .t„ and y^ be the co-ordinates of any point at 
 pleasure, we have 
 
 Distance of point (x^, y^) from line (a, b, c) = ^' " "* •^° "^ ^ , 
 
 Va"^ ->- b* 
 a result already obtained in § 41. 
 
THE STRAIGHT LINE. 61 
 
 These results may be summed up in a third fundamental 
 principle of Analytic Geometry, as follows: 
 
 If 
 
 P ^ ax -\- by -\- c 
 
 be any function of the co-ordinates of the first degree, then — 
 
 I. To every point on the plane will correspond one definite 
 value of P. 
 
 II. T/iis value of P is equal to the perpendicular distance 
 of the point from the line P = multiplied by the constant 
 factor Va' + l)\ 
 
 If the expression P is in the normal form, we have 
 
 «' + ^' = 1, (§ 36) 
 
 and the factor last mentioned becomes unity. 
 Hence — 
 
 III. If we have a function of x and y of the form 
 
 X Q,Q& a -\- y ^m a — p^ Py 
 this function will express the perpendicular distance of the 
 point tvhose co-ordinates are x and y from the line P == 0. 
 
 EXERCISES. 
 
 1. Let the student draw the line 
 
 2^: - Sy + 1 = 0, 
 and let him compute the values of the expression 
 
 2a; - 3?/ + 1 
 for a number of points, and lay them down, as in the scheme 
 of § 51, until he sees clearly the truth of all the preceding 
 conclusions. 
 
 2. Imagine a plane covered with values of the function 
 
 P ^ ax -\- by -\- Cy 
 as in § 51. Around the origin as a centre we describe a circle 
 of arbitrary radius, and on its circumference mark the points 
 where the values of P which it meets are greatest and least. 
 Show that all points thus marked lie on the line bx— ay — 0. 
 Show also on what line the points will fall if the centre of 
 the circle is at the point (^, q). 
 
PLANE ANALYTIC GEOMETRY, 
 
 Theorems of the Intersection of Lines. 
 
 55. We rej)resent by the symbols P, P', etc., Q, Q\ etc., 
 different linear functions of the co-ordinates; e.g., 
 
 P ^ ax -{-by -}- c; 
 
 P' = a'x + h'y -f c'; 
 
 P" = a"x + b''y + c"; 
 
 etc. etc. etc. 
 
 Also, we shall represent by the symbols i/, if', etc., N, N', 
 etc., such functions reduced to the normal form in which 
 
 a" -\-h' = 1. 
 
 Since the Jf's, iV^'s, etc., will be a special case of the 
 P's, §'s, etc., every theorem true of all the latter will also be 
 true of the former; but the reverse will not always be the 
 case. 
 
 The line corresponding to the equation 
 
 P = 
 may, for brevity, be called the line P. 
 
 56. Theorem. If 
 
 P = 0, P' = 
 
 he the equaiions of any tivo straight lines, and if /i and v he 
 any two factors lohich do not contain x or y, then the equa- 
 tion 
 
 ^P J^ypf = (/,) 
 
 will he that of a third straight Ihie passi7ig through the^Joint 
 of intersection of the lines P and P' . 
 
 Proof 1. By substituting in {h) for P and P' the ex- 
 pressions which they represent, we see that jj-P -\- vQ is a 
 function of the first degree in x and y. 
 
 Hence (Z>) is the equation of some straight line. 
 
 2. That point whose co-ordinates satisfy both of the equa- 
 tions P = and P' = must also give yuP -f vP' = 0, 
 and must therefore lie on the line {IS). 
 
THE STBAIOUT LHIK 63 
 
 But such point is the point of intersection of the lines 
 P and P'. 
 
 Hence the point of intersection lies on the line {h), and 
 {h) passes through that point. Q. E. D. 
 
 Corollary. If three functions, P, P' and P", are so re- 
 lated that toe can find three factors, A, yu and v, lohich satisfy 
 the identity 
 
 \P + ywP' + vP" = 0, 
 
 then the three lines P = 0, P' = and P" = intersect 
 ill a point. 
 
 For we derive from this identity 
 
 whence, by the theorem, P passes through the point of inter- 
 section of P' and P" . 
 
 57. Theorem. Conversely, 
 
 If P = 0, P' = and P" = are the eqiiations of three 
 lines intersecting in a point, it aliuays ivill he possible to find 
 three coefficients, p., v and A, such that 
 
 jxP + vP' + AP" E 0. 
 
 Proof. 1. Let the values of the three functions P, P' 
 and P" be 
 
 P ^ax -{- hy -\- c = 0; j 
 
 P' E a'x + h'y + c' = 0; V {a) 
 
 P" = a*'x + V'y + c" = 0. ) 
 
 2. Let us now suppose 
 
 /i E ^"^> - aV', K {b) 
 
 V E ah' — ft'Z*; ) 
 and let us form the expression AP -(- jiP' -f- i^P". In this 
 expression we shall have 
 Coefficient oi x = a(a'V' - a"V) + a'{a"h - ah") 
 
 ^a'\ah' - a'h) (e 0); 
 Coefficient of ?/ = h(ci'h" - a"h') + h\a"h - ah") 
 
 -\-h"{ah' - a'h) (eO); 
 Absolute term = c{a'h" - a"h') -f c\a"h - ah") 
 
 + c"{a'b' - a'h). 
 
64 PLANE ANALYTIC OEOMETRT. 
 
 Because the three lines pass through a point, this absolute 
 term is zero (§ 49). Hence the whole expression is identically 
 zero, and the values {b) of A, ^i and v satisfy the conditions 
 of the theorem. 
 
 EXERCISES. 
 
 1. Show that if the equations P = and § = are so 
 related that we can find two coefficients, fx and r, which form 
 the identity 
 
 ^P-\-vQ = 0, 
 
 then the two lines P and Q are coincident. (Comp. §§52, 53.) 
 
 2. Having the two lines 
 
 y — mx ^ a — 0, 
 
 y + mx -j- 2a = 0, 
 it is required to find the equation of a third line passing 
 through their point of intersection and through the origin. 
 
 Method of Solution. Calling the given expressions equated to zero 
 P and Q, and noting that the equation of every line through the point 
 of intersection may be expressed in the form 
 
 we are to determine the quantities jj- and v so that this line shall pass 
 
 through the origin. Hence the absolute term must vanish. This gives 
 
 the condition 
 
 /I = — 2r, 
 
 the value of v being arbitrary. Substituting this value of//, and divid- 
 ing by y, "we find the required equation, 
 
 y — Zmx = 0. 
 
 3. Find the equation of a line passing through the origin 
 and through the point of intersection of the lines 
 
 y — 2x — a = 0; 
 y ^2x -\- 3a = 0. 
 
 4. Find the equations of the lines making angles of 45° 
 and 135° respectively with the axis of Xand passing through 
 the point of intersection of the above two lines. 
 
 58. To complete and apply the preceding theory, it is 
 necessary to distinguish between the positive and negative 
 sides of a line. If distances measured on one side are positive, 
 those on the other side are negative. But no rule is possible 
 
THE STRAIOET LINE. 65 
 
 for the positive and negative sides without some convention, 
 because the function P may change its sign without changing 
 the position of the line. For example, the two equations 
 
 X — ny -f- 7i = 0, 
 — X -\- ny — h — (d, 
 
 represent the same line; but all values of x and y which make 
 the one function equal to -f P will make the other equal to 
 — P, so that the positive and negative sides of the lines are 
 interchanged by the change of form. 
 
 Now, in the first form, the distance of the origin from the 
 line is 
 
 h 
 
 Hence, 
 
 When the absolute term in the equation is ]JOsUive, the x>osi- 
 
 tive side of the line is that on ^ohich the origioi is situated, and 
 
 vice versa. 
 
 In the normal form the absolute term is negative. Hence, 
 In the normal form a i)0sitive value of the function 
 
 31 = X cos oi -\- y sin a — ]) 
 
 indicates that the point whose co-ordinates are x and y is on 
 the opposite side of the line from the origin , and a 7iegative 
 value that it is on the same side as the origin, 
 
 59. Theorem. If 
 
 if = 0, i\r = 
 
 are the equations of any two lines in the nor7nal form, tlien 
 the equations 
 
 if+JV=0, Jf-iV=0, 
 
 will he the equations of the bisectors of the four angles ivhich 
 the lines Hand Nform at their point of intersection. 
 
 Proof. 1. Because the functions if and i\^are in the nor- 
 mal form, they represent the respective distances of any point 
 from the lines i/ = and iV = 0. (§ 54.) 
 
66 PLANE ANALYTIC GEOMETRY. 
 
 2. Every pair of co-ordinates which fulfil the condition 
 
 M ± N= 
 must give 
 
 M =^ ± N, 
 
 so that the point which they represent is equally distant from 
 the lines M and N. 
 
 3. By geometry, the locus of the point equally distant from 
 two lines is the bisectors of the angles formed by the lines. 
 
 Remark 1. This theorem holds equally true of the equations of 
 auy two lines in which the sums of the squares of the coefficients of x 
 and y are equal. For if, in the equations 
 
 P ^ax -{-by -\-c =0, 
 P^a>x-\- h'y -f c' = 0, 
 
 we have a' + ^* = ^" + ^'^ then, by § 53, the functions Pand P', when 
 not restricted to zero, express the distances of a point {x, y) from the re- 
 spective lines Pand P', multiplied by Va^ -j- b'^ and Va"^ ^ b"^ respec- 
 tively. 
 
 Now, when these multipliers are equal, every point whose co-ordi- 
 nates satisfy the equation 
 
 (d ± a') x + {b ± b') y -\- c ± c' = 
 or 
 
 P± P =0 
 
 must be equally distant from the lines Pand P'. 
 
 Remark 2. The equation 
 
 M - N=0 
 
 will be that of the bisector of the angle in which the origin is situated, 
 and of its opposite angle; while the equation 
 
 M-{-N=0 
 
 will represent the bisector of the two adjacent angles. 
 
 EXERCISES. 
 
 Find the bisectors of the angles formed by the following 
 pairs of lines: 
 
 1. a; _ 2?/ = and 2x — y = 0. 
 
 2. y -\- nx — c = and 717/ — x -\- c = 0. 
 
 3. Prove the theorem of geometry that the two bisectors 
 of the angles formed by a pair of intersecting lines are at 
 right angles to each other. 
 
::::( '«> 
 
 THE STRAIOUT LINE. 67 
 
 In other words, if the functions P and P' are such that 
 cCi + 62 == a'i _|_ j'2^ 
 
 then show that the two lines 
 
 P -f P' = and P - P' = 
 intersect at right angles. 
 
 4. Show that if iV = and N' = are the equations of 
 two lines in the normal form, then 
 
 XN + }iN' = 0, 
 
 XN - }iN' 
 
 will represent the loci of those points whose distances from N 
 and N' are in the ratio // : A. Also, show geometrically that 
 such a locus is a straight line. 
 
 5. In the preceding exercise, what condition must the co- 
 efficients X and yu satisfy in order that the equations {a) may 
 each be in the normal form? 
 
 60. Applications of the Preceding Theorems. The pre- 
 ceding theorems enable us to prove with great elegance the 
 leading theorems of the intersections of certain lines in a tri- 
 angle. 
 
 I. The bisectors of the interior angles of a triangle meet in 
 a 2)oint. 
 
 Proof. Let 
 
 X = 0, if = 0, N=0, 
 be the equations of the sides of the triangle. 
 
 We suppose the origin to be within the triangle, because 
 we can always move it thither by a transformation of co-ordi- 
 nates. 
 
 Then, by what precedes, 
 
 P = L - M=0, 
 
 p' =M- ]sr= 0, 
 
 P"^N- L = 0, 
 
 will be the equations of the bisectors. But these functions, 
 P, P' and P", fulfil the identity 
 
 p _|_ P' + P" = 0, 
 
 and reduce to the form § 56 when we suppose 
 X = jii = V = 1. 
 
PLANE ANALYTIC GEOMETRY. 
 
 Hence P, P' and P" all pass through a point. 
 
 II. The hisedors of any two exterior angles and of the 
 third interior angle meet in a point. 
 
 Proof The equations of two exterior bisectors and of the 
 third interior bisector are 
 
 P ^ L-\- M 
 P' ^M-\- N 
 P"^ L- N 
 
 0; 
 0; 
 0; 
 
 which fulfil the identity 
 
 P - P' 
 
 P" E 0. 
 
 III. Tlie perpendiculars from the three vertices of a tri- 
 angle upon the opposite sides meet in a point. 
 
 Let P be any point upon the perpendicular from Y upon 
 al3; PM X Ya, PN ± F/?; and 
 y = angle aY/3. ^ 
 
 Then, because the angles PYa ^^ 
 
 and a are complementary, 
 
 PM= PY cos a, / PT 
 
 PN = PYcos p] 
 
 PM : PN = cos <a: : cos /?. 
 
 Therefore, if the equations of the sides Ya and Y/3 are 
 
 N = : N = 0, 
 
 then, by the theorem of § 59, Ex. 4, the equation of the per- 
 pendicular YP will be 
 
 N cos p ~ N cos a = 0. 
 
 In the same way, if the equation of ap is N" = 0, we 
 shall have, for the equations of the other two perpendiculars, 
 
 N cos a — N' cos y = 0; 
 N' cos y — N cos /3 = 0. 
 
 The sum of these three equations is identically zero, thus 
 showing that the three lines intersect in a point. 
 
THE STRAIGHT LINE. 
 
 69 
 
 61. Diagonals of a Quadrilateral. An elegant and in- 
 structive application of the preceding theory is given by the 
 following problem: 
 
 To find the cquatioiis of the diagonals of a quadrilateral of 
 luhich the equations of the four sides are given. 
 
 We remark that, in general geometry, a quadrilateral has three 
 diagonals. The reason is that each side is supposed to be of indefinite 
 length, and so to intersect the 
 three others. A diagonal is 
 then defined as the line joining 
 the point of intersection of any 
 two sides to the point of in- 
 tersection of the other two 
 sides. The number of points 
 of intersection, or vertices, is 
 equal to the combinations of 
 two in four, or 6. Taken in 
 pairs these 6 points have three 
 junction lines, as shown in the 
 figure. 
 
 Solution. Let the equations of the four sides be 
 P = 0: 
 
 Q =^\ 
 
 R = 0: 
 S =0.. 
 
 We seek for four factors, k, A, yu and v, by which to form 
 the identity 
 
 kP -{-\Q-\- ^xR^ v8^ 0. {h) 
 
 Four such factors can always be found when the parame- 
 ters of Py Q, etc., are given, because by equating to zero 
 the coefficients of x and g and also the absolute term in (b) 
 we shall have three equations which determine any three of 
 the four factors jc, A, /x and v in terms of the fourth. To 
 the latter we may assign any value at pleasure. 
 
 The identity (b) being satisfied, we shall have 
 
 hP-{-XQ=- (m^ + vS). (c) 
 
 Now, (§ 56), 
 
 H,P -irXQ = 
 
 (a) 
 
70 PLANE ANALYTIC GEOMETRY. 
 
 is the equation of some line passing through the intersection 
 of P and Q, while 
 
 fxR J^ vS =0 
 is the equation of some line passing through the intersection 
 of R and S. 
 
 But, by (c), these two lines are identical. Hence this com- 
 mon line is a diagonal of the quadrilateral. We show in the 
 same way that 
 
 kP ^ lxR = or XQ-\- vS =0 
 is the equation of the diagonal joining the intersection of P 
 and R to that of Q and 8. Also, that 
 
 kP -{- yS =0 or \Q-\- ^R = 
 is the equation of the diagonal joining the intersection of P 
 and S to that of Q and R. 
 
 Example. To find the equations of the diagonals of the 
 quadrilateral whose sides are 
 
 P= X -{- 2/4-1 = 0: 
 
 Q~ a; + 2?/ - 3 = 0: 
 
 R= .T - 2?/ + 4 = 0; 
 
 ^ = 2.^' - ?/ - 2 = 0. 
 
 Forming the expression (Z>), we find it to be 
 
 (;i+;i+/^+2y)a;+(7i+2;i-2//-^)?/ + «-3A+4/^-2rE0. 
 
 Hence, to form this identity, (§ 8), 
 
 (1) ;^ + A + /I 4- 2r = 0; 
 
 
 (3) 
 
 « + 
 
 2A - 
 
 2jx - V 
 
 = 0; 
 
 
 (3) 
 
 H — 
 
 3A 4- 
 
 4// - 2v 
 
 = 0. 
 
 We solve 
 
 as follows: 
 
 
 
 
 
 (3)- 
 
 -(1) 
 
 X - 
 
 3;( - 3v 
 
 = 0; 
 
 
 (3)- 
 
 -(3) 
 
 5A - 
 
 3A 4- 
 
 6/^4- ^ 
 7t^ = 0; 
 
 = 0. 
 
 
 
 
 9/1 4- 16r = 0. 
 
 
 
 
 
 A = 
 
 7 
 -3" = 
 
 21 
 
 
 
 
 fX = 
 
 16 
 
 - ¥ "' 
 
 
 
 
 
 H = 
 
 19 
 
 
THE STllAIOHT LINE. 71 
 
 The value of v is arbitrary, and values of u, pt and A, free 
 from fractions, are obtained by putting y = 9. The values 
 of the four coefficients are then 
 
 ;^ = 19; A = - 21; yu = - 16; t- = 9. 
 
 From these coefficients the equations of the diagonals are 
 formed by the preceding formulae, and are found to be 
 
 2.r + %Zy - 82 = 0; 
 
 x-\-Yty -\h ^ 0; 
 
 37a: + \^y +1 = 0. 
 
 63. Fundamental Lines of a Triangle. Let us consider 
 the following problem: 
 
 If the equations of the three sides of a triangle in the nor- 
 mal form are 
 
 M = 0, 
 
 M" = 0, 
 
 what line is represented hy the equation 
 
 M -\- M' -\- M" = 0? {a) 
 
 Solution, If we put 
 
 Q = M-\- M', 
 
 the equation § = will represent the bisector of the exterior 
 angle between the lines if and M' (§ 59). 
 Also, the equation 
 
 Q^M" = 0, 
 
 which is the same as {a), will represent some line passing 
 through the point of intersection of if" and Q, that is, 
 through the point in which the bisector meets the opposite 
 side. 
 
 In the same way it may be shown that the line («) passes 
 through each of the other two points in which the bisectors 
 of the exterior angles meet the opposite sides. 
 
 Hence the solution of the problem leads to the theorem : 
 TJie three points in which the bisectors of the exterior 
 
72 PLANE ANALYTIC GEOMETRY. 
 
 angles of a triangle meet the op2)Osite sides lie in a straight line, 
 namely, the line lohose equation is 
 
 M-\- M' -^r M" = 0; 
 
 31 = 0, 3r = and M" = beiiig the equations of the sides 
 in the normal form. 
 
 We may show in the same way that the three equations 
 
 M -^ M' - M'' = 0, 
 
 J/ - if' + if" = 0, 
 
 -M + if + M" = 0, 
 
 are the equations of three straight lines each containing the 
 foot of one bisector of an exterior angle and two bisectors of 
 the two remaining interior angles. 
 
 EXERCISES. 
 
 1. Show by the preceding theorems that if we form a 
 triangle by joining the points in which each bisector of an 
 interior angle meets the oi:>posite side, the sides of this tri- 
 angle will severally pass through the points in which the 
 bisectors of the exterior angles meet the opposite sides. 
 
 2. Show that if 
 
 31 = 0, 3r = 0, 3r' = 0, M'" = 0, 
 
 be the equations of the four sides of a quadrilateral in the 
 normal form, then 
 
 3/ _|_ 3f' _|_ J/" -I- if'" = 
 
 will be the equation of a straight line containing the three 
 points in which the external bisectors of the three pairs of 
 opposite vertices meet each other. 
 
 3. Find the equations of the three diagonals of the quad- 
 rilateral whose sides are 
 
 y = x; 
 
 y = X -{-h'y 
 X = a; 
 y = - X. 
 
CHAPTER IV 
 
 THE CIRCLE. 
 
 Section I. Elementary Theory. 
 
 Eqviation of a Circle. 
 
 63. Problem. To find the equation of a circle,'^ 
 Let the co-ordinates of the T 
 
 centre G of the circle be a 
 and i, and let P be any point 
 of the circle. 
 
 Calling X and y the co- 
 ordinates of Py we have, for 
 the square of the distance be- 
 tween G and P, 
 
 CP" = {x- a)' + (y- by. (§ 17) 
 
 The condition that P shall lie on the circle requires that 
 this distance shall be equal to the radius of the circle. Let 
 us put 
 
 r E GP, the radius of the circle. 
 
 The condition then becomes 
 
 (X - ay -\-iy- by = r% (1) 
 
 which is the required equation of the circle. 
 
 64. Theorem. Every equation letween recta7igular co- 
 ordinates of the form 
 
 H^' + f) -{-px + qy-i-h = (2) 
 
 * In the almost universal notation of the higher geometry the word 
 ' ' circle" is used to designate the closed curve which, in elementary 
 geometry, is called the circumference of the circle. 
 
74 PLANE ANALYTIC GEOMETRY. 
 
 in tohich the coefficients of x^ and if are equal, tvhile there is 
 no term in xy, represents a circle. 
 
 Proof. Diyide by m, and put, for brevity, 
 
 
 
 
 a = - 
 
 p 
 
 
 
 
 
 b^ - 
 
 2m' 
 
 
 and the 
 
 equati 
 
 ion will be transformed into 
 
 
 
 
 x' - 2ax ■ 
 
 i-f- 
 
 .2by-{-- = 
 
 0, 
 
 or 
 
 
 
 
 
 
 
 (x- 
 
 ■ «)' + (2/ 
 
 -bf 
 
 - a' -h' ^ 
 
 h 
 
 or 
 
 
 
 
 
 
 
 {X- 
 
 ■ «)' + (y 
 
 -by 
 
 = a' -^b' - 
 
 h 
 
 0, 
 
 (3) 
 
 The first member represents the square of the distance be- 
 tween the fixed point («, b) and the varying point {x, y). The 
 second member being a constant, the equation shows that 
 the square of the distance of the two points is a constant, 
 whence the distance itself is a constant. Hence the equation 
 represents a circle whose centre is at the point (a, b) and 
 
 whose radius is y^^ \ y^ _ z^ 
 
 m' 
 
 65. Special Forms of the Equation oj a Circle, 
 
 We may suppose a circle moved so that its centre shall 
 
 occupy any required position without the form or magnitude 
 
 of the circle being changed. 
 
 If the centre be at the origin, we have a = and ^ = 0, 
 
 and the equation of the circle becomes 
 
 x^ -Vy' = r\ (4) 
 
 If the centre is on the axis of X, we have Z> = 0, and the 
 equation becomes 
 
 f^{x- ay = r\ 
 
 which is the equation of a circle whose centre is on the axis 
 of X. 
 
THE CIRCLE. 
 
 75 
 
 If we suppose a = r and 5 = 0, the circle will be tangent 
 to the axis of Y at the origin, and the y 
 equation will become 
 
 = 2ax — x^, (5) 
 
 which we may define as the equation 
 of a circle when a diameter is taken as 
 the axis of X and the origin is at the 
 end of this diameter. 
 
 EXERCISES. 
 
 Find the radii and the co-ordinates of the centres of circles 
 having the following equations : 
 
 1. x" + 2/' - lO.r 4- 2y + 17 = 0. 
 
 2. 2,x' + 3?/' + Qx - 12?/ - 9 = 0. 
 
 3. 2^:^ + 2if + 8a; - 18?/ - I = 0. 
 
 4. mx^ + my'' + P^'^ + ([V 
 
 2m 
 
 5. Write the equation of the circle whose centre is at the 
 point (1, — 2) and whose radius is 7. 
 
 6. Write the equation of the circle whose centre is in the 
 position {p, q) and whose radius is Vp^ + q". 
 
 7. Write the equation of the circle whose centre is at the 
 point (0, 5) and which is tangent to the axis of X. 
 
 8. Write the equation of a circle passing through the 
 origin and having its centre at the point (3, 4). 
 
 9. Find the equation of a circle of which the line drawn 
 from the origin to the point {p, q) shall be a diameter. 
 
 10. Find the equation of a circle of which the line from 
 the point (1, 3) to the point (7, — 5) shall be a diameter. 
 
 11. Find the locus of the centre of the circle passing 
 through the points {p, q) and (p', q'), and show that it is a 
 straight line perpendicular to the line joining these points. 
 
76 PLANE ANALYTIC GEOMETRY, 
 
 MetJiod of Solution. Since the two points are to lie on the circle, their 
 co-ordinates must satisfy the equation of the circle; that is, we must 
 have 
 
 (P - af + iq -6)2 = r^ 
 
 ip' - ay + iq' - hf = r\ 
 The radius r being a quantity which must not appear in the equation, we 
 must eliminate it, which we do by mere subtraction. We thus find au 
 equation of the first degree between a and b, the co-ordinates of the 
 centre. To express the locus in the usual form we may write x and y 
 for a and h in this equation, which will then be the required equation of 
 the locus of the centre. 
 
 12. Find the locus of the centre of the circle passing 
 through the points (1, 1) and (7, 9). 
 
 13. Find the locus of the centre of the circle passing 
 through the origin and the point (p, q). 
 
 14. Find the locus of the centre of the circle passing 
 through the origin and the point {2p cos a, 2p sin a). 
 
 66. Intersections of Circles. The points in which circles 
 intersect each other, or in Avhich a straight line intersects a 
 circle, are found from the values of the co-ordinates which 
 satisfy both equations. 
 
 Let the two circles which intersect be given by the equa- 
 tions 
 
 x' -{-f -i-ax -j-hij -^p =0;) , . 
 
 ^' + 2/' + «'^' + ^'.y + ^/ = 0. f ^ ^ 
 
 By subtracting one of these equations from the other, we 
 have 
 
 (« - a')x + (b- h')y + ;j - y = 0; 
 
 whence y = ^ /~ ^ —, —. 
 
 By substituting this value of y in either of the equations 
 (a), we shall have a quadratic equation in x. 
 
 Since such an equation has two roots, there will be two 
 points of intersection. 
 
 But the roots may be imaginary. The circles will then 
 not meet at all, but one will be wholly within or wholly with- 
 out the other. 
 
 If the roots are equal, the points of intersection are coinci- 
 dent, and the circles touch each other. 
 
THE CIRCLE. 77 
 
 EXERCISES. 
 
 1. Find the points of intersection and the length of the 
 common chord of the two circles 
 
 a:' + I/' - rta: = d\ 
 
 Ans, The co-ordinates are: 
 
 r^ — d"^ 
 X = for both points; 
 
 a 
 Common chord = - | a'r'- (r' - dy \ * 
 
 2. Find the points of intersection and tlie length of the 
 common chord of the circles 
 
 x" -\-y'' -a'^ 0; 
 x''^f-\-by -7-^ = 0, 
 
 3. Determine the radius 7' so that the circles 
 
 x" -Yy"" -2x = 3, 
 
 shall touch each other. 
 
 MetJiod of Solution. We find, as in the preceding exercises, the 
 values of the co-ordinates x and y of the points of intersection. In order 
 that the roots may be equal, the quantity under the radical sign in the 
 expression for y must vanish. Equating it to zero, we shall have 
 
 7^ _ 107'-' + 9 = 0, 
 
 an equation of which the roots are 3 and 1. 
 
 4. Find the distance apart of the two points in which the 
 line 
 
 x = y-\-l 
 intersects the circle 
 
 x" -{-y' = 10. 
 
78 PLANE ANALYTIC GEOMETRY. 
 
 67. Polar Equation to the 
 Circle. 
 
 Let be the pole, OX the 
 initial or base line; 
 
 p' and a the i)olar co-ordi- 
 nates of the centre C; 
 
 p and 6 the polar co-ordi- 
 nates of any point P. 
 
 We then have, by trigonom- o' 
 
 etry, 
 
 PC = OF' + OC - 20P. OC cos FOG; 
 that is, r' = p' + p" — 2pp' cos {d — a), 
 or p^ - 2pp' cos {6- a)-\- p" - r^ = 0, (6) 
 
 the polar equation required. 
 
 It may also be obtained from the equation referred to 
 rectangular axes by putting x = pcosO, y = p sin 0, a = p' 
 cos ex, and b = p' sin a. If the initial line pass through the 
 centre, a = and the equation becomes 
 
 p' - %pp' cos ^ 4- p'= - r= = 0. (7) 
 
 If the origin lie on the circumference, p'' = r' and the 
 equation becomes 
 
 p = 2p' cos ^ = 2r cos 0, (8) 
 
 Note. In the above we put p and p' for the radii vectores in order 
 to avoid confusing them -with the radius of the circle, which we call r. 
 
 Tangents and Normals. 
 
 68. Equation of Tangent to a Circle. The requirement 
 that a line shall be tangent to a circle does not alone determine 
 the line, because a circle may have any number of tangents. 
 We may therefore anticipate that this requirement will be ex- 
 pressed by an equation of condition between the parameters 
 of the line. Let us then consider the problem: 
 
 To find the equation -which the parameters of a liiie nmst 
 satisfy in order that the line may he tangeiit to a given circle. 
 Let the circle be given by the equation 
 (X - ay +{y- by = r\ 
 and let the equation of the line be 
 
 Ax + % + C = 0. 
 
THE CIRCLE. 79 
 
 By geometry, the situation of the line must be such tliat 
 the perpendicular from the centre of the circle upon it shall 
 be equal to the radius of the circle. Conversely, every line 
 for which this perpendicular is equal to the radius of the 
 circle is a tangent. 
 
 Now, the length of the perpendicular from the point {a, b) 
 upon the line {A, B, 0) is 
 
 aA-^hB-^ Q 
 VA^~+~B~' 
 
 The requirement that this perpendicular shall be equal to 
 the radius r of the circle gives the equation 
 
 aA + bB-}- C=rVA' + B% (1) 
 
 which is the required equation of condition between the para- 
 meters A, B and C. 
 
 If the equation of the line is in the normal form 
 xcos a -{- y sin a — 2:^ = 0, 
 we shall have 
 
 VA' -\-B' = Vcos'a + sin^a: = ± 1, 
 and the equation (1) will assume the form 
 
 acos a -{- b sin a — 2^ = ± r, (2) 
 
 69o Equation of the tangent expressed in terms of the tan- 
 gent of the angle tuhich the line mahes ivith the axis of X. 
 Let y — mx -J- 5 be the equation of the tangent, and 
 
 the equation of the circle. Eliminating y between these two 
 equations, we have 
 
 (1 + m^)x^ + 2mbx + {W - r') = 0, 
 which must have equal roots, since the tangent touches the 
 circle in only one point. Now the condition that this equa- 
 tion may have equal roots is (§ 8) 
 
 m'b' = (l-\-m') {b' -r'); 
 whence b = r Vl + m% 
 
 which substituted in the equation of the tangent gives 
 
 y = mx ^r Vl -\- m^ (3) 
 
80 PLANE ANALYTIC GEOMETRY. 
 
 Conversely, every line whose equation is of this form is a 
 tangent to the circle. 
 
 70. Tangent determined by Tivo Conditions. In the 
 preceding article we employed only the one condition, that the 
 line should be tangent to the circle. Hence the line could 
 be completely found only when one of the parameters was 
 given. In order to determine completely the tangent line, 
 some other condition besides its tangency to the circle must 
 be given. Examples of such conditions are: 
 
 That the tangent line shall touch the circle at a given 
 point; 
 
 That it shall pass through a given point not on the circle ; 
 That it shall also be tangent to a second circle. 
 
 71. Problem. To fi^id the equation of the line tangent 
 to a circle and passing through a given point. 
 
 Let ic' and y' be the co-ordinates of the given point, and 
 X cos OL -^ y sin a — p =^ 
 the equation of the tangent. Since the tangent passes through 
 the point {x', y'), we must have 
 
 x' cos a -\- y' sm a — p = 0, (4) 
 
 which combined with (2) will determine the two parameters, 
 a and p, of the line. 
 
 We may, however, first eliminate p by subtraction, wdiich 
 gives the equation 
 
 {a — x') cos a -{- {h — y') sin a = r. 
 
 The solution of this equation, which is obtained by 
 
 methods given in trigonometry, will give the value of a. It 
 
 may also be obtained algebraically by substituting for cos a 
 
 its equivalent, Vl — sTnP a, or for sin a its equivalent, 
 
 Vl — cos"" a, or for cos a and sin a their equivalents in 
 
 terms of tan a, viz., 
 
 1 . tan a 
 
 cos a = - =: sm a — 
 
 Vl + tan^a VI + tan^ 
 
 In either case we shall have a quadratic equation, the un- 
 known quantity in which will be either sin a, cos a or tan a. 
 
THE CIRCLE. 
 
 81 
 
 If we write, for brevity, 
 
 VI = a — a:', 
 n = b -if, 
 
 the solution of these equations will give 
 
 sm a = 
 
 cos a 
 
 tan a 
 
 nr ± m, Vm"^ -\- n"^ — r^ 
 
 
 =F 
 
 m' -\- 
 
 if 
 
 
 mr 
 
 n Vm"" 
 
 -\-n^ 
 
 -r' 
 
 
 ± 
 
 m' + 
 
 n' 
 
 
 mn 
 
 r Vm' 
 
 + n^ 
 
 -r' 
 
 (5) 
 
 r — n 
 The value of j9 by (4) is 
 
 p z=. x' cos a -\- y' sin a, 
 
 in which cos a and sin a must be replaced by their values 
 given above. Substituting the values of cos a, sin a and ;:> 
 in the equation 
 
 X cos OL -\- y sin a: — ^j = 0, 
 
 the result will be the required equation of the tangent passing 
 through the point (x\ y'). The double sign shows that there 
 may be tioo tangents drawn to a circle from a point without 
 it. 
 
 Case when the given point is on the circle. In this case we 
 shall have 
 
 rrv" + n" - r"" = 0, 
 
 and the values (5) of sin a and cos a become 
 
 sm a = -; 
 r 
 
 cos a 
 
 m 
 
 and 
 
 p = 
 
 mx* 4- '^y* 
 
 Substituting these values in the equation of the tangent, 
 it becomes 
 
 mx -\- ny — mx' — ny* = 0, 
 or 771 (x — x') -\- n (y — y') = 0. 
 
82 
 
 PLANE ANALYTIC GEOMETRY. 
 
 If we substitute for m and n their values, the equation is 
 
 {a - x') {X - X') ^{b- y') (y - y') = 0. 
 
 If we take the centre of the circle as the origin, we have 
 
 a = Q, b = 0; 
 
 and because the point (x', y') is on the circle, 
 
 x'' + y'' = r\ 
 
 Making these substitutions, the equation of the tangent 
 assumes the simple form 
 
 x'x + y'y = r\ (6) 
 
 72. Def. The subtangent of a curve is the projection 
 on the axis of X of that portion of the tangent intercepted 
 between the point of tangency and its intersection with the 
 axis of X. 
 
 Thus, if CT is the axis of 
 X, MT is the subtangent corre- 
 sponding to the tangent PT. 
 
 Length of the Subtangent. To 
 find the length of MT, we find 
 the intercept GT on the axis of 
 X and subtract CM, the abscissa 
 of the point of contact. 
 
 The equation of the tangent P^is (6) 
 
 x'x -\- y'y = r'; 
 and when ?/ = 0, we have 
 
 r 
 
 X — ~ 
 
 X* 
 
 Hence we have 
 
 CT. 
 
 MT = 
 
 r — X 
 
 X = - 
 
 X X 
 
 73. Def. The normal to any curve is the perpen- 
 dicular to the tangent at the point of contact. 
 
 Equation of the Normal to a Circle. The equation of the 
 line perpendicular to 
 
 x'x -|- y*y = r"^ 
 
THE CIRCLE. 83 
 
 and passing through the point of contact {x^, y') is, by §47, 
 
 7/' 
 
 y -y' ^ '^A^ - ^'). 
 
 or y'x — x'y = 0, 
 
 the equation required. 
 
 The form of this equation shows that every normal of a 
 circle passes through the centre — a property which is easily 
 established by elementary plane geometry. 
 
 The length of the normal is that portion of the line in- 
 cluded between the point of contact and the axis on which the 
 subtangent is measured. In the case of the circle, the nor- 
 mal is constant and equal to the radius. 
 
 Def. The subnormal of a curve is the projection of 
 the normal on the axis of X. 
 
 Thus, C3I is the subnormal corresponding to the point P, 
 and in the circle is equal to the abscissa of the point of contact. 
 
 EXERCISES. 
 
 1. Show that the condition that the line 
 X cos a -\- y sin a — p = 
 shall be tangent to the circle 
 
 (x-ay-j-(y-by = a'+b'^ 
 
 is a cos a -{- b s'm a =p ± Va^ -j- b'\ 
 
 2. What is the condition that the line 
 
 y = mx -\- c ^ 
 shall be tangent to the circle c 5? -(>«*+ ^)-4/V^T7T>i?>P^ 
 
 (X - ir + (2/ + 2y = 16? 
 
 3. What must be the value of c in order that the equation 
 
 x + y = c 
 
 may be tangent to the same circle? 
 
 Ans. c= -1 ±4:V2. 
 
84 PLANE ANALYTIC GEOMETRY. 
 
 4. What must be the value of 711 in order that the line 
 
 y = mx + 6 
 may be tangent to the circle 
 
 x" -\-y' = 16? 
 
 2"* 
 
 5. In the last example show that we get the same an- 
 swer for the line 
 
 y = mx — 6, 
 
 and explain the equality by geometric construction. 
 
 6. What must be the value of the radius d in order that 
 the circle 
 
 (X + 3)' + (2/ - ir = d^ 
 
 may have as a tangent the line 
 
 ^ = 22: + 5? 
 
 Ans. d = — -zr. 
 
 7. By elementary geometry, two circles are tangent to each 
 other when the distance of their centres is equal to the sum 
 or difference of their radii. By means of this theorem write 
 out the condition that the circles 
 
 {x-aY -{-{y-hy =r' ;(^-^7N-'^*'~-Vr 
 
 and {x-ay-^(y-by = r'' - ^ ±t^ 
 
 shall be tangent to each other. 
 
 8. Show that the length of the common chord of the circles 
 whose equations are 
 
 (X - 2Y ^{y-3Y = 9 
 and (x - 3)'' + (?/ - 2)^ = 9 
 
 is VU. 
 
 9. Find the condition that tlie circles 
 
 {x - hy + (y- ky = a' 
 and {x - hy + (2/ - hy = a' 
 
 may touch each other. 
 
 Ans. a = —-=. (Jr. — h). 
 
THE CIRCLE. 85 
 
 10. Show that the polar equation 
 
 p' — {a cos 6 -\-b sin 0) p = p^ 
 
 is that of a circle, and express its radius and the position of 
 its centre. 
 
 11. What curve does 
 
 p z= a C0& (6 — a) -\- b cos {0 — /3) -\- c cos (^ — k) + • • • 
 represent? 
 
 12. A point moves so that the sum of the squares of its 
 distances from the four sides of a rectangle is constant. Show 
 that the locus of the point is a circle. 
 
 13. Given the base of a triangle {2b) and tlie sum of tliQ 
 squares on its sides (2m^), find tlie locus of the vertex when 
 the middle point of the base is the origin. 
 
 Ans. x" -f ?/^ =1 11 f — W. 
 
 14. Given the base ifi) and the vertical angle [B) of a tri- 
 angle, find the locus of the vertex when the origin is at the 
 end of the base. 
 
 Ans. x^ -\- if — bx — by cot B — ^. 
 
 15. Show that if, in the equation 
 
 x'^J^ifJ^Ax-\-By^C^^, 
 we have 
 
 4(7 > A^ ^B\ 
 
 the circle will be imaginary. It is enough to show that the 
 radius is imaginary. 
 
 16. Show that a circle may be defined as the locus of a 
 point the square of whose distance from a fixed point is pro- 
 portional to its distance from a fixed line. 
 
 17. Show that a circle is the locus of a point the sum of 
 the squares of whose distances from any number of fixed 
 points is a constant. 
 
 18. If -- = 77, show that the circles 
 
 a' b' 
 
 x^ -\- y"^ -\- ax -\-by =0, 
 
 x^ + 2/' + «'^ + ^'y =- 0, 
 
 touch at the origin. 
 
86 PLANE ANALYTIC QEOMETRT. 
 
 19. Find the locus of a point wliose distances from two 
 fixed points have a fixed ratio to each other. 
 "- 20. Express analytically the locus of a point from which a 
 tangent drawn to a circle will have a fixed length t. 
 ^^ 21. Find the locus of the point from which two adjoining 
 segments of the same straight line shall be seen under the 
 same varying angle. In other words, if A, B and C are 
 three points in the same straight line, find the locus of the 
 point X which will satisfy the condition 
 
 I Angle AXB = angle BXC. 
 
 >f\j 
 
 ^ 22^ If the equation of the circle x^ -{- y"^ = 
 to another system of co-ordinates having the same ax^ but a 
 different direction, show analytically that the equation will 
 not be altered. 
 
 23. Show analytically that if a circle cuts out equal chords 
 from the tv/o co-ordinate axes, the co-ordinates a and b of its 
 centre will be equal. t^M^x^ur «w-c «. ^ «/ r »■- a '- "y * > Jt^-cl-^ 
 
 24. Find the equation of the circle which passes through 
 the three fixed points (x^y^), {x^ij^), {x^y^). 
 
 25. Having given the circle x^ + ^^ + ^^x — 6y — 2 = 0, 
 find the equation of its two tangents, each of which is parallel 
 to the straight line y = 2x — 7. y ^ 2 ^ -/- ,j ±u^y 
 
 26. The circle x"^ -{- y^ = r"^ has tangents touching it at the 
 respective points {x^ yj and (x^ y^. Express the tangent of 
 the angle formed by these tangents. ^I/x' "/--?':_ 
 
 27. A line of fixed length slides along the axes of coordi- 
 nates in such a way that one end constantly remains on each 
 axis. What is the locus of the middle point of the line? 
 
 28. Given a point {a, h) and a finite straight line whose 
 length is c, find the locus of the point whose distance from 
 {a, h) is a mean proportional between c and its distance from 
 the line x cos ol -\- y sin a ^= p. 
 
 ■ 29. Having given the equation of the circle y"^ = 2rx — x^, 
 let chords be drawn from the origin to all points of the circle, 
 and let each of them be divided in the constant ratio m : n. 
 It is required to find the locus of the points of division. 
 
 30. The same thing being supposed, the chords, instead 
 
THE CIRCLE. 87 
 
 of being divided, are each doubled. Find the locus of the 
 ends. 
 
 31. On each radius of a circle having its centre at the 
 origin a distance from the origin is measured equal to the or- 
 dinate of the terminal point of the radius on the circle. Find 
 the locus of the point where the measures end. ^^^y^ - ^ ^ 
 
 32. The same thing being supposed, take on each radius 
 a point at a distance from the centre equal to the abscissa of 
 
 the end of the radius, and find the locus of this point, a V ^ V /jc^ 
 
 33. Find the locus of the point from which two circles p'^z," 
 will subtend the same angle; that is, from which the angle ^ y 
 subtended by the pair of tangents to one circle is equal to 
 that subtended by the pair of tangents to the other circle. 
 
 34. Find the equation of that circle which passes througli 
 the origin and cuts off the respective intercepts J9 and q from 
 
 the positive parts of the axes of JTand Y. (x -|j2- -/■ c^~ \)^- r"^ 
 
 35. Find the locus of the point the sum of the squares of 
 whose distances from the sides of an equilateral triangle is I 
 constant, and show that it is a circle. (To simplify the prob- / w 
 lem, let the base of the triangle be the axis of X.) i^^^^^y^^-aQ v-^ia'i 
 
 36. Find the polar equation of the circle when the origin 
 
 is on the circumference and the initial line a tangent. P= it" i^9- 
 
 37. A line moves so that the sum of the perpendiculars 
 AP and BQ from two fixed points, A and B, shall be a con- 
 stant. Find the locus of the middle point of the segment >>^ 
 
 pQ. :tvr= c^-t^r ^JL: 
 
 38. The straight line whose equation is 3^ + 5rc -|- 19 == ^ ^ 
 cuts the circle y"" -\- x" = 113 in two points. Wliat is the r*-/^ 
 length of the chord which the circle cuts off from the line? - //j-2i 
 
 39. Find the equation of the straight line which cuts the 
 circle x^ -\- y^ = 169 in two points whose abscissa3 are re- 
 spectively — 12 and -f 7. 
 
 40. Find the equation of a line passing through the point 
 {x\ y') and forming in the circle x" -\- y"" = r'' o, chord whose 
 length is J. -^c^^^+^x^.^-/. =o > = /r- eL^ ^'vv^^-a^'j^^-^^o 
 
 41. Through the point (x', ?/'), inside the same circle, a 
 chord is to be drawn which shall be bisected by the point. 
 Find the equation of the chord. 
 
 /^ 
 
 
88 PLANE ANALYTIC QEOMETBT. 
 
 Systems of Circles. 
 
 *74:. Let us consider the expression 
 
 {X - ay -^r{y- bf - d\ 
 which, for brevity, we shall represent by P, putting 
 P ^x" -{-y' -'Hax- 2by -^ a" -\- V - d'^ 
 
 To every point on the plane will correspond a definite 
 value of P, found by substituting the co-ordinates of such 
 point in this value of P, 
 
 We may form any number of expressions of this form, 
 such as 
 
 P' ^x' -{-y' - 2a' X - Wy + a'"" + V - d'^-, 
 
 P" ^x' ^y' ~ 2a"x - V)"y + a"'' + V" - d'"; 
 
 etc. etc. etc. 
 
 In general, the co-ordinates x and y which enter into P 
 will be considered as entirely unrestricted, in which case P 
 will be simply an algebraic function of x and y. 
 
 But we may also inquire about those special values of x 
 and y which satisfy the equation P — 0. AVe know from 
 §§ 63, 64 that the points corresponding to these special 
 values of x and y all lie on a circle of radius d, having its 
 centre at the point {a, h). We now have the — 
 
 75. Theorem. 77^6 value of P for any jyoint of the plane 
 is equal to tlie square of the tangent from that point to the 
 circle P = 0. 
 
 Proof Let P be the point {x, y), ^^-r- 
 and let O be the centre of the circle 
 P = 0, which is, by hypothesis, the 
 point (a, h). We then have 
 
 Cr = {x- ay + (^ - hy-, 
 and because PTC is a right angle, 
 PT' = GP' - CT' 
 
 = (^ - ay +(2/ - by - d\ 
 
 :^p 
 
THE CIRCLE. 89 
 
 which is the vahie of the function P, thus proving the 
 theorem. 
 
 Remark. If the point {x, y) is taken within the circle, P 
 will be negative, and the length of the tangent, being the 
 square root of P, will be imaginary. 
 
 76. Theorem. If P = o and P' = are the eq^iaiions 
 of two circles, any equatmi of the form 
 
 ^P + vP' = {a) 
 
 will represent a third circle passing through their points of 
 intersection. 
 
 Proof. We first show that {a) is the equation of some 
 circle. Substituting for P and P' their values, we have for 
 the equation of the curve 
 
 {}^-\-y){x'' + f) - 2{Ma + va')x - 2(//J + vy)y 
 
 + /^(a^ -\-b' -d')-\- y{a" + h'' - d'') = 0. 
 
 Here the coefficients of x^ and y^ are equal and there is no 
 term in xy. Hence (§ 64) the curve represented by the 
 equation {a) is some circle. 
 
 Secondly, the co-ordinates of all points in which the 
 circles P and P' intersect must satisfy both of the equations 
 P = and P' = 0. Hence they also satisfy the equation 
 
 //P + rP' = 0, 
 
 and therefore the points of intersection lie on the circle of 
 which the equation is {a). 
 
 Hence this circle passes through the points of intersection 
 of the circles P and P'. Q. E. D. 
 
 Cor. The curve represented by (a) depends only on the 
 ratio of the factors jj. and v, and remains unchanged when 
 both are multiplied by the same quantity. 
 
 By assigning different values to the ratio /< : v, we may 
 determine as many circles as we please passing through two 
 points. 
 
 A collection of circles passing through two points is called 
 a family of circles. 
 
90 PLANE ANALYTIC OEOMETRT. 
 
 77. Problem. To find the locus of tlie -point from loliich 
 the tangents to tiuo circles shall have a given ratio to each 
 other. 
 
 Solution. Let P = and P' = be the equations of the 
 two circles, and let the tangents from the moving point be in 
 the ratio m : m'. 
 
 The square of the tangents will then be in the ratio 
 m^ : m'^. But these squares are represented by the respective 
 values of P and P' corresponding to the point from which 
 the tangents are drawn. Hence between these values of P 
 and P' we have the proportion 
 
 P: P' = nf : m'\ 
 which gives 
 
 m''P - m'P' = 0. (b) 
 
 Because the co-ordinates of the point from which the 
 tangents are drawn must satisfy this equation, this equation 
 is that of the required locus. 
 
 Comparing with § 76, we see that the equation is of the 
 form (a). Hence: 
 
 Theorem. T7te locus of the point from lohich the lengths 
 of the tangents drawn to tivo circles have a constant ratio to 
 each other is a third circle, passing through the common points 
 of intersection of the first two circles, and therefore a third 
 circle of the same family. 
 
 78. Tlie Radical Axis. If the ratio m : m' is unity, the 
 equation {h) will reduce to 
 
 P - P' = 0, 
 
 or, substituting for P and P' their values, 
 
 2(a' - a)x + 2(&' - h)y + a" - a' + h'' -V-d'^^d''^ 0, 
 
 which, being of the first degree, is the equation of a straight 
 line. From the results of § 76, this line must be the common 
 chord of the two circles. Hence: 
 
 Theorem. Ilie lociis of the point from which the tangents 
 to two circles are equal is the common chord of the two circles. 
 
 Tliis locus is called the radical axis of the two circles. 
 
THE CIRCLE, 91 
 
 Imaginary Points of Intersection. 
 
 79. Tlie tliC(:>rcni of § 77 holds equally true wlieilicr 
 the circles F and P' intersect or not; that is, it leads to a 
 third circle passing through the points of intersection of two 
 circles, even 2uhe7i these two circles do not intersect. If in this 
 last case the third circle, which we may call P", really inter- 
 sected either of the others, say P', this result would be self- 
 contradictory. For in such a case the circle P could not pass 
 through the intersection of P' and P", and so the result of 
 the theorem would be false. 
 
 But if the point of intersection of P and P' is nowlwre, 
 there will be nothing contradictory in the result, if only P" 
 intersects each of them nowhere. 
 
 Again, in § 78 we have found a perfectly general equation 
 of the radical axis founded on the definition that the radical 
 axis is the line joining the points of intersection of two circles, 
 •which equation gives the real radical axis even when the circles 
 do not intersect. 
 
 If we take any special case, w^e shall find that the algebraic 
 processes are the same whether the circles do or do not really 
 intersect: only, in the latter case, the co-ordinates of the 
 points of intersection will be imaginary. To illustrate this in 
 the simplest way, take the two circles 
 
 {x - 3)^ + (?/ - 3)^ = 9. 
 To determine the points of intersection we must find 
 values of x and y which satisfy both equations. The second 
 equation is, by reduction, 
 
 x'-\-if -Qx- G?/ + 18 = 9. (a) 
 
 Substituting the value of x^ + t/' = 1 in the first member, 
 we find 
 
 10 5 
 
 Hence 
 
 , 5 , , 10 , 25 
 
92 PLANE ANALYTIC QEOMETRT. 
 
 By substituting for ?/' its value 1 — x", we find 
 
 Completing square. 
 
 2 1/ O 
 
 ^ -3 ^=-9- 
 
 
 i-e, 
 
 , 5 25 25-32 
 "^ 3^"^36~ 36 ~ 
 
 7 
 36' 
 
 The square being negative shows that the roots are imagi- 
 nary. The solution gives, for the points of intersection, 
 
 5 ± V^^ 
 
 X — 
 
 y 
 
 6 
 
 5 qp 4/- 
 
 The co-ordinates x and y being imaginary, the circles do 
 not really intersect. But these imaginary values of the co- 
 ordinates satisfy the equations of both circles and also the 
 equation [b) of the radical axis, as we readily find by the 
 calculation : 
 
 5^ 
 
 - 7 ± 10 V^ 
 
 • 7 
 
 9 ± 5 V- 
 
 ■ 7 
 
 5^ 
 
 38 
 
 _ 7 If: 10 |/Z: 
 
 ~7 
 
 18 
 9 T 5 V- 
 18 
 5 
 ~3 
 
 ? 
 ^ 
 
 5 
 
 36 
 
 ±7 + 5:f 7 
 6 
 
 10 
 ~ 6" 
 
 ) 
 
 2/^ 
 x-^y 
 
 In taking the sum of the first two equations, the imaginary 
 terms cancel each other and we have x^ -{- y"^ = 1. 
 
 Subtracting 6 times the third equation we satisfy {a), and 
 the third is identical with (5), which is the equation of the 
 radical axis. 
 
 We adopt the following forms of language to meet this 
 class of cases: 
 
 I. An imaginary point is a fictitious point which we 
 sup2?ose or imagine to be represented by imaginary co-ordi- 
 nates. 
 
 II. When imaginary co-ordinates satisfy the equation of a 
 
THE CIRCLE. 93 
 
 curve, we may talk about the corresponding imaginary points 
 as belonging to that curve. 
 
 III. A curve may be entirely imaginary. 
 
 Example. The equjition 
 
 x" -{-if -2x-2y=\ -3 
 is that of a circle. But we may write it in the form 
 
 {X - ly + (2/ - 1)' = - 1. 
 
 The first member is a sum of two squares, and therefore 
 positive for all real values of x and y, while the second mem- 
 ber is negative. Hence there are no real points whose co- 
 ordinates satisfy the equation. 
 
 EXERCISES. 
 
 1. If we take, on the axis of X, two imaginary points ^ 
 whose abscissas are a -f- hi and a — hi respectively, find the 
 abscissa of the middle point between them. 
 
 2. Using the method of § 45, find the equation of the 
 line joining the imaginary points whose co-ordinates are — 
 
 1st point: x' — ci, y' ~ a -\- 2ci; 
 
 2d point: x/' = h -\- ci, y" = a -{-2h + 2ci; 
 
 and show that it is the real line 
 
 y = 2x -^ a. 
 
 3. Find the equation of the circle whose centre is at the 
 point («, 2^) and which cuts the axis of Xat the points de- 
 scribed in Ex. 1. 
 
 A71S. (x - ay + (y - 2hy = U\ 
 
 4. Find the equation of a circle belonging to the family 
 fixed by the pair 
 
 (x-%y + (y-5Y= 9, 
 and passing through the origin. 
 Ans, •/< = 20; v=- 9; Eq. : ll(a;^-f ?/') - 124a: - 30y = 0. 
 
94 
 
 PLANE ANALYTIC QEOMETBT. 
 
 Section II. Synthetic Geometry of the Circle. 
 
 Poles and Polars. 
 
 80. Let there be two points, P and P', on the same 
 straight line from the centre of a fixed circle, and so situ- 
 
 ated that the radius OR shall be a mean proportional between 
 OP and OP'. 
 
 Through either of the points, as P', draw a line Q perpen- 
 dicular to the radius. Then 
 
 The line Q is called the polar of the point P with 
 respect to the circle, and the point P is called the pole of the 
 line Q with respect to the circle. 
 
 Had we drawn the line through P, it would have been the 
 polar of the point P', and P' would have been the pole of the 
 line through P. 
 
 81. The following propositions respecting poles and 
 polars flow from these definitions: 
 
 I. To every point in the plane of the circle corresponds 
 one definite polar, and to every line one definite pole. 
 
 II. The polar of a point and the pole of a line may be 
 found by construction as follows: 
 
 io) If the pole P is given, we draw the radius through 
 the pole intersecting the circle at R. We then find the point 
 P' by the proportion OP : OE = OR : OP'. 
 
 The perpendicular through P' will be the polar of P. 
 
 (b) If the polar is given, wc draw the perpendicular from 
 
TlIE CIRCLE. 
 
 95 
 
 the centre upon the polar, and produce it if necessary. If 
 it intersects the circle at R and the polar at P', we determine 
 OP as the third proportional to OP' and OR. The point P 
 will then be the required pole. 
 
 III. When the pole is within the circle, the polar is wholly 
 without it. 
 
 IV. If a pole is without the circle, the polar cuts the circle. 
 
 V. When the pole is a point on the circle, the polar is the 
 tangent at that point. 
 
 ■ VI. If the pole ajiproaches indefinitely near the centre of 
 the circle, the polar recedes indefinitely, and vice versa. 
 
 82. Fundamental Theorem. If a line jkiss through a 
 point, the polar of the point ivillpass through the pole of the line. 
 
 Proof. Let the line CD pass 
 through the point P. 
 
 By definition, we find the polar 
 of P by drawing the radius OM 
 through P, taking the point P' so 
 that, putting r for the radius OM, 
 
 OP '.r = r: OP', {a) 
 aod drawing P' Q' \_ 0P\ 
 
 We find the pole of CD by draw- 
 ing OQ 1_ CD and finding a point P' 
 
 such that 
 
 OQ:r = r'. OP". (b) 
 
 We have to prove that P" lies on the polar P'Q'. If we 
 call Q' the point in which OQ meets the polar P'Q', the tri- 
 angles P'OQ' and QOP, being both right-angled and having 
 the angle at common, are equiangular and therefore similar. 
 
 Hence 
 
 OQ: 0P= OP' : OQ'. (r) 
 
 Comparing the proportions {a) and (h), we have 
 
 OP.OP' = OQ.OP", 
 
 which gives the proportion 
 
 OQ: 0P= OP' : OP". 
 
 Comparing this proportion with (c), Ave have 
 
 OQ' = OP". 
 
96 
 
 PLANE ANALYTIC OEOMETRT. 
 
 Hence P" and Q' coincide; that is, the pole P" lies on the 
 pohirP'(2'. Q. E. D. 
 
 Cor, 1. We may imagine several lines all passing, like CD, 
 through the point F. The theorem shows that the poles of 
 these lines all lie on P'Q\ Hence, 
 
 If several lilies j^^ass through a point, their poles luill all lie 
 upton the polar of the point. 
 
 Cor. 2. We may imagine several points, all lying, like P, 
 on the line CD. The theorem shows that the polars of these 
 points will all pass through Q', the pole of CD. Hence, 
 
 If several points lie in a straight line, their polars will all 
 pass through the pole of the line. 
 
 Kemark. These several theorems may be more readily 
 grasped when placed in the following form: 
 
 1. If a line turn round on a point, its pole will move along 
 the polar of that point. 
 
 %. If a p>oi7it move along a line, its polar will turn round 
 on the ptole of that line. 
 
 83. Theorem I. If from any 
 poi7it tivo tangeiits he draiun to a 
 circle, the line joining the points 
 of contact will he the polar of the 
 pdhit. 
 
 Proof. Let the tangents from 
 P touch the circle at M and iV. 
 Let Q be the point in which OP, 
 from the centre 0, intersects the line i/iV. 
 
THE CIRCLE. 97 
 
 By elementary geometry, OQN and ONP are right tri- 
 angles. Because they have the angle at common^ they are 
 equiangular and similar. Hence 
 
 OQ: 0N= ON: OP, 
 
 Now, since OiVis the radius of the circle, this proportion 
 shows that MNh the polar of P. Q. E. D. 
 
 Theorem II. If through any point a chord he drawn to 
 a circle, the tangents at the extremities of the chord will meet 
 071 the polar of the point. 
 
 Proof. Let the chord pass 
 through the point Q, and let 
 the tangents meet at P'. By 
 Theorem I. , P' is the pole of the 
 chord; therefore, because Q lies 
 on the chord, the polar of Q 
 passes through P', the pole of 
 the chord. Q. E. D. 
 
 Cor, 1. If any number of chords le drawn through the 
 same point, the locus of the point in which the tangents at 
 their extremities intersect will be a straight line, the polar of 
 the point. 
 
 Cor. 2. Conversely, If from a moving point on a straight 
 line tangents be draivn to a fixed circle, the chords joining the 
 corresponding points of tangency will all pass through the pole 
 of the line. 
 
 THEOREMS FOR EXERCISE. 
 
 1. If we take any four points, A, B, A' and B', on a circle, 
 and if P be the point of meeting of the tangents at A and B, 
 and P' the point of meeting of the tangents at A' and B% 
 then the point of meeting of the lines AB and A^B' will be 
 the pole of PP'. 
 
 2. If we take four points. A, B, Xand T, on a circle, such 
 that the tangents at A and B and the secant XT pass through 
 a point, then the tangents at Xand l^and the secant AB 
 will also pass through a point. 
 
98 PLANE ANALYTIC GEOMETRY. 
 
 Centres of Similitude. 
 
 84. Def. The line joining the centres of two circles is 
 called their central line. 
 
 Theorem. If the ends of parallel radii of two circles he 
 joined ly straight lines, these lines tuill all 2^(^ss through a 
 common point on the central line. 
 
 Proof. Let GP and C'P' be any two parallel radii, and 
 8 the point in which the line PP' intersects the line CC 
 joining the centres. The similar triangles SPG, SP'G' give 
 the proportion 
 
 8G : SG' = GP : G'P'. (a) 
 
 Putting, for brevity, r = the radius GP, and r' ~ the radius 
 C'P', this proportion gives, by division, 
 
 SG - SG' : SG' = GP - G'P' : G'P' = r- r' :r\ 
 or GG' : SG' = r- r' : r'. 
 
 Ilcnce SG' =-^,GG'; 
 
 r — r 
 
 that is, the distance SG' is equal to the line GG' multiplied 
 by a factor which is independent of the direction of the radii 
 GPf G'P; therefore the point S is the same for all pairs of 
 parallel radii. Q. E. D. 
 
 Gase of oppositely directed radii. If the radii CP, G'P' 
 be drawn in opposite directions, it may be shown in a similar 
 
THE CIRCLE. 99 
 
 way that the line PP' intersects the central line CC in a 
 point 8' determined by the proportion 
 
 CS' : C'S' = CP : C'P' = r : r', {b) 
 
 whence S' is a fixed point in this case also. 
 
 Def. The two points through which pass all lines joining 
 the ends of parallel radii of two circles are called the cen- 
 tres of similitude of the two circles. 
 
 The direct centre of similitude is that determined 
 by similarly directed radii. 
 
 The inverse centre of similitude is that determined 
 by oppositely directed radii. 
 
 Corollaries. The following corollaries should, so far as 
 necessary, be demonstrated by the student. 
 
 I. The direct centre of sitliilitude is ahvays ivitliout the 
 central line of the two circles, and the inverse centre is always 
 loithin this line, hotuever the two circles may he situated. 
 
 II. If the tioo circles are entirely external to each other, the 
 centres of similitude are the points of meeti^ig of the pairs of 
 common tangents to the two circles. 
 
 III. Comparing the proportions {a) and {h), we see that the 
 point S divides the line CC^ externally into segments having 
 the ratio r : r', while aS" divides it internally into segments 
 having this same ratio. 
 
 This is the definition of a harmonic division. Hence 
 The two centres of similitude divide harmonically the li?ie 
 joining the centres of the tiuo circles. 
 
 85. The following are fundamental theorems relating to 
 centres of similitude: 
 
 Theorem I. Every line similarly dividing tiuo 2)arallel 
 radii of two circles passes through their centre of similitude. 
 
100 PLANE ANALYTIC GEOMETRY. 
 
 Proof. Let A and A' be the points in which the line di- 
 vides the radii; 
 
 S'f the point in which this line cuts the central line; 
 S, the centre of similitude; 
 r, r', the radii of the circles. 
 
 We then have, by the property of the centre of similitude, 
 
 a 8 : CO' = C'P' : CP - C'P' = r' :r-r\ {a) 
 The similar triangles CAS' and C'A'S' give 
 C'8' : CS' = C'A' : CA; 
 whence, by division, 
 
 C'S' : CC = C'A' : CA - C'A\ (h) 
 
 By hypothesis, the radii are similarly divided at A and A'; 
 
 CA' : CA = r' :r; 
 whence, by division, 
 
 CA' : CA - CA' = r\r-r'. 
 Comparing with {a) and (b), 
 
 CS' : CC = CS : CC; 
 whence CS' = CS and the points S and S' coincide. Q.E. D. 
 
 Kemark 1. If the radii are oppositely directed, the 
 centre of similitude will be the inverse one. The demonstra- 
 tion is the same in principle. 
 
 Eemark 2. The demonstration may be shortened by 
 employing the theorem of geometry that there is only one 
 
THE CIRCLE. -101 
 
 point, internal or externiil, in which a line can be divided in 
 a given ratio. (See Elementary Geometry.) By the funda- 
 mental property of the centre of similitude, it divides the 
 central line into segments proportional to the radii of the 
 circles. It may be shown that the point ;S" divides the central 
 line into segments proportional to CA and C'A'. From this 
 the student may frame the demonstration as an exercise. 
 
 Theorem II. Conversely, If any line pass through a 
 centre of similitude, and j^ctt^^Uds he drawn from the centres 
 of the circles to this line, the lengths of these parallels loill he 
 proportional to the radii of the circles. 
 
 The demonstration is so easy that it may be supplied by 
 the student. 
 
 86. The Four Axes of Similitude of Three Circles. If 
 there be three circles, they form three pairs, each with its 
 direct and inverse centre of similitude. There will there- 
 fore be six such centres in all, three direct ones and three in- 
 verse ones. The following propositions relate to this case: 
 
 Theorem III. The three direct centres of similitude lie 
 in a straight line. 
 
 Proof. Let r^, r, and r^ be the radii of the three circles. 
 
 Let 8^, S^ and 8^ be the direct centres of similitude of the 
 pairs of circles (2, 3), (3, 1), (1, 2) respectively. 
 
 Let a line AB hQ passed through 8^ and 8^. 
 
 From the centres of the three circles draw three parallel 
 lines, i?j, R^ and B.^ to the line AB. Then, 
 
 Because AB is a line passing through the centre of simili- 
 tude 8^, 
 
lora 
 
 PLANE ANALYTIC GEOMETRY. 
 
 B, :B, = r,: r,. (Th. 11.) 
 
 Because AB isn line passing through /S,, 
 
 B, : B, = r, : r,. 
 Taking the quotients of these ratios, we have 
 
 B,:B,= i\ : r,; 
 hence the line AB divides the radii r, and 1\ similarly. 
 
 Therefore this line passes through the centre of similitude 
 8, of the circles (1, 2). (Th. I.) Q. E. D. 
 
 Theorem IV. Each direct centre of similitude lies in t^ie 
 same line with the tiuo inverse centres of similitude lohich are 
 7iot paired luith it. 
 
 The demonstration of this theorem is so nearly like that 
 of the last one that it may be supplied by the student. 
 
 Def. A straight line which contains three centres of simili- 
 tude of a system of three circles is called an axis of simili- 
 tude of the system. 
 
 Corollary. For each system of tliree circles there are four 
 axes of similitude, of which one contains the three direct cen- 
 tres of similitude, and the others each contain one direct and 
 two inverse centres. 
 
 EXERCISES. 
 
 1. Show that if two of the three circles be equal, two of 
 the axes of similitude will be parallel, and vice versa, 
 
 2. If all three circles are equal, describe the axes of simili- 
 tude. 
 
THE CIRCLE. 
 
 103 
 
 The Radical Axis. 
 
 87. Theorem. If any 'perpendicular ho drawn to the 
 ceiitral line of tivo circles, the difference of the squares of the 
 tangents from any one point of this perpendicular ivill be the 
 same as from every other point of it. 
 
 Proof. Let PiV^ be any per- 
 pend iculiir to the central line 
 of the circles C and 6", and 
 P any point on this perpen- 
 dicular. 
 
 Let R and R' represent the 
 distances of P from C and C\ 
 
 Because the tangents PT 
 and PT' meet the radii drawn 
 to the points T and T of con- 
 tact at right angles, we have 
 
 PT 
 
 -prpn 
 
 R' -r'; 
 R'' - r'\ 
 
 Hence, for the difference of the squares of the tangents, 
 
 PT^ _ PT'^ Z3 i^^ - R'^ - (r^ - r''). (1) 
 
 From the right triangles PNG and PNC, we find, in the 
 same way, 
 
 R^ - R'' = NC - NC"; 
 
 whence, from (1), 
 
 PT^ _ prpn ^ jy-^2 _ j^fjn _ ^^.2 _ ^«) ^^j 
 
 The second member of this equation has the same value at 
 whatever point on the perpendicular P may be situated, 
 which proves the theorem. 
 
 Corollary. If we choose the point i\^so as to fulfil the 
 condition 
 
 NC 
 
 we shall have 
 
 NO" = r' 
 
 p/TT2 J) 'Jin 
 
 (3). 
 
104 PLANE ANALYTIC GEOMETRY 
 
 and the tangents will be equal from every point of the per- 
 pendicular, which will then, by definition, be the radical axis. 
 
 SS, Case luhen the circles intersect. In this case the 
 tangents drawn from either point of intersection are both 
 zero and therefore equal. Hence this point is on the radical 
 axis, and this axis is then the common chord (or secant) of 
 the two circles. Hence another definition : 
 
 The radical axis of two circles is their common chord, 
 produced indefinitely in both directions. 
 
 EXERCISE. 
 
 In the preceding construction the circles have been drawn 
 comi^letely outside of each other. Let the student extend 
 the general proof (1) to the case when the circles intersect, 
 showing that the two tangents from every point of the com- 
 mon secant are equal, and (2) to the case when one circle is 
 wholly within the other, showing that the radical axis is then 
 wholly without the outer circle. 
 
 89. The Radical Centre of 
 Three Circles. If we have three 
 circles, each of the three pairs 
 will have its radical axis. We 
 now have the theorem: 
 
 The three radical axes of three 
 circles intersect in a x^oint. 
 
 Proof Let^, B and C be the 
 three circles, and let be tlie 
 
 point in which the radical axis of A and B intersects 
 radical axis of B and G. 
 
 Because is on the radical axis of A and B, 
 
 Tangent to ^ = tangent to B. 
 Because is on the radical axis of B and C, 
 
 Tangent io B = tangent to C. 
 
 Hence Tangent to ^ = tangent to (7; 
 
 whence lies on the radical axis of A and C, and all three 
 axes pass through 0. 
 
THE CIRCLE. 
 
 105 
 
 Def. Tlie point in which the three radical axes intersect is 
 called the radical centre of the tliree circles. 
 
 Cor, Tlie radical centre of three circles is a certai7i point 
 from which the tangents to the three circles are all equal. 
 
 90. System of Circles having a Common Radical Axis* 
 The theory of a family of circles, developed analytically in the 
 preceding section, will now be explained synthetically. 
 
 Peoblem. Let us have a circle A and a straight line N: 
 it is required to find a second circle X, such that N shall be 
 the radical axis of the circles A and X. 
 
 Solution. From the centre A draw an indefinite line AX 
 perpendicular to the line N. 
 
 Take any point P on the radical axis N, and from it draw 
 a tangent PT to the given circle. 
 
 From the same point, P, draw another line, PT, in any 
 direction whatever; make PT' = PT, and from T' draw T'X 
 perpendicular to PT' and meeting the central line in X. 
 The circle round the centre X with the radius XT' will be 
 that required. 
 
 For PT', being perpendicular to the radius, is tangent 
 to the circle X; and because PT = PT', the line through 
 P perpendicular to the central line is the radical axis. 
 Hence the given line iVis the radical axis of the two circles; 
 whence the circle JT fulfils the condition of the problem. 
 
 Since the line PT' may be drawn in any direction what- 
 ever, we may find an indefinite number of circles which fulfil 
 the conditions of the problem. 
 
10(5 PLANE ANALYTIC OEOMETRT. 
 
 The construction of these circles is shown in the figure. 
 Since the tangents from P are all equal, it follows that the 
 
 line PN is the radical axis of any two circles of a family 
 passing through the same two points, real or imaginary. 
 
 Tangent Circles. 
 
 91. The following propositions lead to the solution of 
 the noted problem of drawing a circle tangent to three given 
 circles. 
 
 Def. When two circles each touch a third, the line 
 through the points of tangency is called the chord of con- 
 tact. 
 
 When two circles touch each other, either one must be 
 wholly within the other, ot each must be wholly without the 
 other. Hence contacts are said to be of two kinds, Mo'^nal 
 and external. 
 
 Theorem I. First, If a circle is tmlgent to a pair of 
 other circles, the chord of co7itact passes through a centre of 
 similitude of the j)air. 
 
 Secondly, This centre of similitude is the direct one when 
 
THE CIRCLE. 107 
 
 the contacts are of the same hind, and the inverse one ivhen 
 they are of opposite kinds. 
 
 Proof. The points of con- 
 tact are readily shown to be cen- 
 tres of similitude of the respective 
 pairs of tangent circles. 
 
 By § 86, any two centres of 
 similitude of different pairs lie 
 on a straight line with one of the 
 centres of similitude of the third pair. 
 
 Hence the points of tangency are in the same line with a 
 centre of similitude. Q. E. D. 
 
 Remark 1. An independent proof of the theorem is ob- 
 tained by drawing the radii from each centre of the pair of 
 circles to the points in which the joining line intersects the 
 circumferences, and showing that the radii, taken two and 
 two, are parallel. 
 
 Remaek 2. The second part of the theorem is left as an 
 exercise for the student. 
 
 02. Homologous Points. If a common secant to two 
 circles be drawn through either of their centres of similitude, 
 it will intersect each circle in two points. By combining 
 
 either of these points on one circle with either of the points 
 on the other circle we may form four pairs of points, as 
 (P, P'). {Q, Q')^ (ft n, and (P, Q'). The pairs at the 
 termini of parallel radii, namely, (P, P') and (Q, Q'), are 
 called homologous points; those at the termini of non- 
 parallel radii, as ((>, P') and (P, Q'), are called anti-ho- 
 mologous. 
 
108 
 
 PLANE ANALYTIC GEOMETRY. 
 
 93. Theorem II. If tivo secants he drmvn through a 
 centre of similittide, then — 
 
 I. Tlie distances of any two homologous points on one secant 
 from the centre of similitude ivill he proportional to the dis- 
 tances of the corresponding points on the other secant. 
 
 II. The products of the distances of two anti-homologous 
 2mnts loill he the same on the tiuo secants. 
 
 Hypothesis. Two secants, SQ' and ST', from the centre 
 of similitude S, cut the circles in the points P, Q, P' and 
 C, and R, T, R' and T respectively. 
 Conclusions : 
 
 I. SP : SP' = SR : SR'; 
 SQ: SQ' = ST: ST. 
 11. SP . SQ' = SR . Sr = SQ . SP' = ST. SR'. 
 
 Proof. I. Draw the central line and the radii to tlie 
 points of intersection. Because of the parallelism of the radii 
 OP and OP', etc., we have 
 
 Triangle SOP similar to triangle SO'P' 
 Triangle SOQ similar to triangle SO'Q'; 
 Triangle SOR similar to triangle SO'R'] 
 Triangle SOT similar to triangle SO'T'. 
 
 From the similarity of these triangles, we have 
 
 SO : SO' --= SP : SP' = SQ : SQ' 
 
 = SR : SR' = ST : ST'. Q. E. D. 
 
 II. The second and last of these proportions give 
 
 SP . SQ' = SP' . SQ; 
 SR . ST' = SR' . ST. 
 
 i\ 
 
 (") 
 
THE CIRCLE. 109 
 
 By a fundamental property of the circle, shown in elemen- 
 tary geometry, 
 
 SP . SQ = SR . ST; 
 
 SF' . SQ' = SR' . sr. 
 
 Multiplying these equations, we have 
 
 SP . SQ' X SP' . SQ = SR. sr X SR' . ST. 
 By substitution from (a), this equation becomes 
 
 (SP . ^'S')' = (^'^ . ^^')'; 
 
 whence, extracting the square root and combining with (a), 
 we have conclusion II. Q. E. D. 
 
 94. Pef, When a circle touches two others, we call it a 
 direct tangency when the two tangencies are of the same kind, 
 and an inverse tangency when they are of opposite kinds. 
 
 Several pairs of tangencies, all direct or all inverse, may 
 be called of the same nature. If one pair is direct and another 
 inverse, they are of opposite natures. 
 
 Eemakk. It will be noted that the chords of contact 
 pass through the same centre of similitude in the case of two 
 pairs of tangencies of the same nature, but not otherwise. 
 Hence, in what follows, whenever we have several circles 
 touching two others, we shall suppose the tangencies to be of 
 the same nature. 
 
 95. Theorem III. //' each circle of one pair is a tan- 
 gent of the same nature to the two circles of another pair, then 
 the radical axis of each pair passes tltrough a centre of simili- 
 tude of the other pair. 
 
 Proof. Let the circles P and P' touch the circles and 
 0' at the points M, N, M' and N'. 
 
 The point of meeting, S, of the lines iVif and N'M' will 
 be a centre of similitude of and 0' (§91). Hence we have 
 
 SM. SN = SM' . SN'. (§ 93) 
 
 But SM . >S'iVis equal to the square of the tangent from S 
 to the circle P (EL Geom.), and SM'. SN' is the square of the 
 
110 PLANE ANALYTIC GEOMETRY. 
 
 tangent from 8 to tlie circle P'. The tangents being equal,. 
 S is on the radical axis of P and P\ Q. E. D. 
 
 It is shown in the same way that a centre of similitude of 
 P and P' is on the radical axis of and 0\ Q. E. D. 
 
 Cor. 1. If each of three circles is a tangent of the same 
 nature to two other circles, then, by this theorem, one of the 
 centres of similitude of each two out of the three circles must 
 lie on the radical axis of the two circles which they touch. 
 Hence, 
 
 When each of two circles touches each of three other circles^ 
 their radical axis tvillform one of the axes of similitude of the 
 three circles. 
 
 Cor. 2. The same thing being supposed, each radical 
 axis of the three circles will, by the theorem, pass through a 
 centre of similitude of the pair which they touch. This centre 
 of similitude will therefore be their point of intersection. 
 Hence, 
 
 W7ie7i each of two circles touches each of three other circles, 
 the radical centre of the three circles will I?e a centre of simili- 
 tude of the two circles. 
 
 96. Problem. To draw a circle tangent to three given 
 circles. 
 
 Construction. Let L, TIf and Wbe the three circles. 
 Find their radical centre, (7, and an axis of similitude, S, 
 
TEE CIRCLE. 
 
 Ill 
 
 Find the poles p, g, r, of 8 with respect to the three 
 circles. 
 
 Join Cp, Cq and 6V, and let ?, ??i, 71 and Z', 7?i', n' be the 
 points in which these lines intersect the three circles. 
 
 The circle through the three points l, m, n will be one of 
 the tangent circles required, and the circle through the three 
 points Vy m', n' will be the other. 
 
 Proof. The axis of similitude 8 is the radical axis of 
 some pair of circles touching the three given circles (§ 95, 
 Cor. 1), and C is one of their centres of similitude (§ 95, 
 Cor. 2). 
 
 Let us call X and Y the two circles of this pair, which are 
 not represented in the figure.* 
 
 Let m, I, n and m', n' , V, instead of being defined by the 
 above construction, be defined as the points of tangency of 
 this pair of circles whose centre of similitude is at C. Then, 
 by (§ 91), the lines mm', nn' and IV will all pass through C 
 
 Through m' draw the common tangent to the circles M 
 and X, and through m draw the common tangent to J/ and 
 Y, and let P be the point of meeting of these tangents. 
 
 * The tangent circles and tangents are omitted from the printed 
 figure to avoid confusing it. The student can supply them so far as 
 necessary. 
 
112 PLANE ANALYTIC GEOMETRY. 
 
 Then, because the tangents Pm and Pm/ touch the same 
 circle M at m and m', they are equal. 
 
 Hence these lines are also equal tangents to the circles 
 X and Y] hence P lies on the radical axis of Xand Y, that 
 is, on the line S (§11). 
 
 Because the line mm' is the chord of contact of tangents 
 from Py it is the polar of P; hence the pole of S, a line through 
 X, lies on the polar mm'. That is, the line Cq, found by the 
 construction, passes through the points of contact m and m'. 
 
 In the same way it is shown that the points of contact I, V 
 and n, n' are upon the lines joining C and the poles j9 and r. 
 Q. E. D. 
 
CHAPTER V. 
 THE PARABOLA. 
 
 Equation of the Parabola. 
 
 97. Def. A parabola is the locus of a jooint which 
 moves ill a plane in such a way that its distances from a fixed 
 point and from a fixed straight line in that plane are equal. 
 
 The fixed point is called the focus, and the fixed straight 
 line the directrix of the parabola. 
 
 The curve is traced mechanically as follows : 
 
 Let F be the fixed point or focus, and RR' the fixed straight line or 
 directrix. Along the latter place the edge of a 
 ruler, and to the focus attach one end of a 
 thread whose length is equal to that of a second 
 ruler, DQ, right-angled at D. Then having at- 
 tached the other end of the thread to the ruler 
 at Q, stretch the thread tightly against the edge 
 of the ruler DQ with the point of a pencil, 
 while the ruler is moved on its edge BR along 
 the directrix RR' : the path of P will be a 
 parabola. For in every position we shall 
 
 have 
 
 PF=PD, 
 
 wiiich agrees with the definition. 
 
 98. Problem. To find the equation of tlie 'parabola. 
 
 Let i^ be the focus, and YY' the directrix. Through i^ 
 draw OX perpendicular to YY', Take y 
 as the origin, OF as the axis of X, 
 and the directrix 01^ as the axis of Y, 
 Put OF = p, and let P be any point on 
 the curve. Join PF, and draw PiV^ per- 
 pendicular to the directrix YY\ Then, 
 by the definition of the curve, we have 
 
 PF=^PN. 
 
114 
 
 PLANE ANALYTIC OEOMETRY. 
 
 Let OM, FM, the co-ordinates of P, be x and y. Then we 
 
 have 
 
 PM' + FM 
 
 that is, 
 or 
 
 y' 
 
 = FF' 
 = FN' 
 = 03P; 
 
 + {x-py = x\ 
 
 y' = 2p 
 
 _ 1 
 
 Ph 
 
 (1) 
 
 which is the equation of the parabola 
 with the assumed origin and axes. 
 
 When ?/ = we have x = ip; that is, OA = AF, or the 
 curve bisects the perpendicular distance between the focus 
 and the directrix; and since there is no limit to the possible 
 distance of a point from both focus and directrix, the curve 
 extends out to infinity. From (1) we see that for every posi- 
 tive value of X greater than p there are two values of «/, equal 
 in magnitude but of opposite signs. Hence the curve is 
 symmetrical with respect to the axis of X, If x be negative 
 or less than -Jjt?, the values of y are imaginary; therefore no 
 part of the curve lies to the left of A. 
 
 Def, The point A where the curve intersects the perpen- 
 dicular from the focus on the directrix is called the vertex 
 and ^Xthe axis of the parabola. 
 
 The equation (1) will assume a simpler and more useful 
 form by transferring the origin to the vertex, which is done 
 by simply writing x for a; — -J/?; hence (1) becomes 
 
 f = '2px, (2) 
 
 a parabola which we 
 Y 
 
 / 
 
 which is the form of the equation of 
 shall use hereafter. 
 
 In equation (2), let x = ^p. 
 Then y^ =i?% 
 
 or y = ± p. 
 
 Hence FL = FU = 2AF, 
 and LU = 2p =: ^AF, 
 
 Def. The double ordinate through 
 the focus is called the principal para- 
 meter or latus rectum. 
 
 Cor. The length of the semi-parameter is p. 
 
 L 
 
THE PARABOLA. 
 
 115 
 
 99. Focal Distance of any Point on the Parabola, 
 
 Let r denote the focal distance FP of any point P (§ 98). 
 Then, by the definition of the curve, we have 
 FP = JVP 
 
 = OA + AM, 
 or r= i^ + X, (3) 
 
 which, being of the first degree, is sometimes called the lin- 
 ear equation of the parabola. 
 
 100. Polar Equation of the Parabola. 
 
 Problem. To find the polar equation of the parabola, 
 the focus being the pole. 
 
 Let FP = r, XFP = 6. Then, from the figure, we have 
 FP = PJV 
 
 = OF-\-FM 
 = 2AF-{-FM, 
 or r =p -{- r cos 6; 
 
 whence r 
 
 P 
 
 1 — cos 6^ 
 
 2 sin' 
 
 6' 
 
 2 
 
 If we count the angle 6 from the 
 vertex in the direction AP, we shall 
 have AFP = 6, and therefore (4) be- 
 comes P _ P 
 
 
 1 + cos 
 
 2 cos^ ^ 
 
 (5) 
 
 which is the form of the polar equation generally used. 
 
 Cor. The polar equation may also be easily deduced from 
 the linear equation of the curve. Thus, when the vertex is 
 the origin, the linear equation is 
 
 r^ip-\- x; 
 and transferring the origin to the focus by writing x -\- ^p for 
 X, it becomes 
 
 r = p -\- X 
 = p — r cos 0; 
 
 whence -- — 
 
 as before. 
 
 1 + cos & 
 
116 
 
 PLANE ANALYTIC OEOMETRT. 
 
 Diameters of a Parabola. 
 
 101, Def. The diameter of a parabola is the locus of 
 the middle points of any system of parallel chords. 
 
 Pkoblem. To find the equation of any diameter. 
 
 Let X, y be the co-ordinates of P, the middle point of any 
 chord CG'\ x' , y' the co-ordinates of C"; r = PC, half the 
 length of the chord; 6 the inclination of 
 the chord to the axis of the curve. Draw 
 the ordinates PM, CM' and PD parallel 
 to AM'. Then we have 
 
 AM' = AM ^ PD, 
 
 or x' :=i X -\- r cos d, 
 
 and CM' = PM-\-C'D, 
 
 or y^ = y -\- r sin d; 
 
 and since the point (x', y') is on 
 
 curve, we have 
 
 y'' = 2px', {c) 
 
 Substituting the values of x' and y' as given by (<^) and 
 {h) in (c), we have 
 
 {y -j- r sin Oy = 2p(x -f r cos 0), 
 or r' sin'^ + 2{y sin 6 — p cos 6)r + ^^ — 2px = 0, 
 from which tb determine the two values of r. But since the 
 point (x, y) is the middle of the chord, the values of r are 
 equal in magnitude but of opposite signs; therefore the co- 
 efficient of r must vanish, which gives 
 
 y sin 6 — p cos = 0, 
 or y = P cot 6 
 
 -P. 
 
 where m = tan 6, the slope of the chord to the axis of X, 
 
 Hence the equation of any diameter is 
 
 y =pcote. ^ (6) 
 
 Since the second member of (6) is constant for any system 
 of parallel chords, evei^y diameter of a 2Mrabola is a straight 
 
THE PARABOLA. 117 
 
 line parallel to the axis o/X(§40, III.). Because m may have 
 any value whatever, (6) can be made to represent any straight 
 line parallel to the axis of the curve. Hence every line paral- 
 lel to the axis bisects a system of parallel chords. 
 
 Cor. To draw a diameter of the curve, bisect any two 
 parallel chords, join the points of bisection and produce the 
 line to meet the curve: it will be a diameter. 
 
 Tangents and Normals. 
 
 102. Problem. To find the equation of a tangent to a 
 parabola. 
 
 Let (a;', ?/') and (:c", y") be the co-ordinates of any two 
 points on the curve. Then the equation of the secant through 
 these points is 
 
 y-y' = P^(^ - =^')- («) 
 
 But since (a;', ?/') and {a;", ?/") are on the curve, we have 
 
 «/" = 2px' (b) 
 
 and 2/"" = 2i?a;". (c) 
 
 From (b) and (c) we get 
 
 ^'- - /» = 2pix^^ - xy, 
 
 whence ^„ ~ ^, = -^ — ., 
 
 which substituted in (a) gives, for the equation of the secant, 
 
 y-y' = 7^^(^ - ^')- 
 
 Now when the point (x'^, ?/") coincides with the point 
 (a;', y'), the secant will become a tangent, and then a;" = a;' 
 and 2/" = y^; hence the equation of the tangent at the point 
 {x', y') is 
 
 y -f ^ p-(^ - ^% 0) 
 
 or y'y = i^x — p)x^ + y" 
 
 = px — px* 4- 2px* 
 = P(x + x'). (8) 
 
118 PLANE ANALYTIC GEOMETRY. 
 
 Cor. Let {x', y') be the co-ordinates of any point on a 
 parabola; the equation of the tangent at that point is 
 
 y=P-(x + x'), («) 
 
 and the equation of the diameter passing through the point 
 C< y') is 
 
 Eliminating if from (a) and (J), we have 
 
 y — m{x -\- ic') 
 for the equation of the tangent. But m is the slope of the 
 parallel chords to the axis; hence the tangent at the extremity 
 of any diameter of a parabola is parallel to the chords ivhich 
 are bisected by that diameter. 
 
 The equation of the tangent may also be derived in- 
 dependently of the point of contact in the following manner. 
 
 103. PiiOBLEM. To find the conditio7i that the line 
 y = mx -\- h 
 may be tangent to a given parabola. 
 The equation of the curve is 
 
 whence, by eliminating y between these equations, we get 
 
 {mx -\-hy = 2px, 
 or nv'x'' + {27nh - 2p)x + h' = 0, 
 
 which determines the abscissae x of the two points in which 
 the line intersects the curve. But since the line is to be a 
 tangent, the two values of the abscissae will be equal. The 
 condition that this equation may have equal roots* is 
 {27nh - 2p)y = 4.7}fh'; 
 
 whence h = - — , 
 
 2m 
 
 * The condition that the roots of the quadratic equation 
 ax"^ + &a; + c = 
 shall be equal is 6^ — iac = 0. (Chap. I.) 
 
THE PABABOLA. 
 
 119 
 
 the required condition. Substituting this value of h in the 
 equation of the line, we have 
 
 y = mx -j- - 
 
 P 
 
 2m' 
 
 (0) 
 
 which is the equation of the tangent to a parabola in terms of 
 the slope and semi-parameter. 
 
 Conversely, every line whose equation is of this form is a 
 tangent to a parabola. 
 
 104. The SuUangent. Def. The subtangent is the 
 projection of the tangent upon the axis of the parabola. 
 
 To find where the tangent meets the axis of X, make 
 2/ = 0, in (8), and we get 
 
 = ^(^ + x'), 
 or . X = — x'-, 
 
 that is, AT=AM', 
 
 or, the suUangent is bisected at the rp, 
 vertex. 
 
 This property enables us to draw 
 a tangent at any point on a para- 
 bola. Thus, let P be any point on 
 the curve; draw the ordinate PM, and produce 3IA to T, 
 making AT equal to A3f; join TP. Then TP is the tangent 
 required. 
 
 105. The Normal. Def. The normal to a curve at any 
 point is the perpendicular to the tangent at that point. 
 
 Problem. To find the equation of the normal to a loara- 
 tola. 
 
 The equation of the tangent at any point {x! , y') has been 
 shown to be 
 
 y 
 
 j{x^xy. 
 
 and let 
 
 y — y' ^z m{x — x') 
 
 (J) 
 
 be the equation of a line through (a:', ?/') and normal to the 
 curve at that point. 
 
 Now in order that the lines represented by (a) and (h) 
 
120 PLANE ANALYTIC OEOMETBT. 
 
 may be perpendicular to each other, we must have the con- 
 dition 
 
 ^' + 1-0, (§47) 
 
 or m = — ^; 
 
 P 
 
 therefore {h) becomes 
 
 y-y' = -j^^-«^')' (10) 
 
 which is the equation of the normal at the point (x', y'). 
 
 106. The Subnormal. Def. The subnormal is the 
 
 projection of the normal upon the axis of the parabola. 
 
 To find where the normal intersects the axis of X, make 
 2/ = in (10). Then we have 
 
 p =1 X — x' 
 =AN-AM 
 
 that is, the suhnormal MN is coiistant and equal to half the 
 pammeter or latus rectwn. 
 
 107. Theorem. A tangent to a parabola is equally in- 
 clined to the axis of the curve and the focal line from the point 
 of tangency. 
 
 Proof From (§ 104) we have 
 
 FT= AT-\- AF 
 
 = AM-i- AF 
 = FF; 
 
 therefore the angle PTF is equal to the angle FPT. 
 
 Cor, Let PD be drawn parallel to the axis AX; then PD 
 is a diameter of the curve (§101), and the angle IIPI) is equal 
 to the angle TPF. Since the normal PJY is perpendicular to 
 the tangent, the angle DPN is equal to the angle NPF, 
 
 Remark. The properties just proved find an application in the use 
 of parabolic reflectors intended to bring rays of light to a focus, as in 
 the reflecting telescope. Since the curve and the tangent have the same 
 direction at the point of tangency, rays of light are reflected by the 
 
THE PARABOLA. 121 
 
 curve as they would be by the taugeut at that point; and because the 
 angle of incidence is equal to the angle of reflection, it follows that if 
 rays of light parallel to the axis of the curve fall upon a parabolic reflec- 
 tor, they will all be reflected to the focus. Conversely, if a luminous 
 body be placed in the focus of a parabolic reflector, all the rays proceed- 
 ing therefrom will be parallel after reflection. 
 
 108. Problem. To find the locus of the foot of the per- 
 pendicular from the focus upou a variable tangent. 
 
 Let x', if be the co-ordinates of any point P on the curve. 
 The equation of the tangent at P is 
 
 The equation of the line through the focus whose co-ordi- 
 nates are (^j, 0), and perpendicular to {ct), is 
 
 And since the point {x\ y*) is on the curve 
 
 y'' = 2px'. (c) 
 
 we have now to eliminate x' and y' from (a), (b) and (c). 
 From (c), 
 
 x' -^ 
 -2p' 
 
 which substituted in (a) gives 
 
 From {b) we have 
 
 y'- py 
 
 X - ip 
 which substituted in (d) gives, after obvious reductions, 
 
 {f + (^ - ipy\x = 0. 
 
 Therefore we must have either 
 
 y^ + (x - ipy= or a: = 0. 
 
 The former gives ^ = and x = ^;, the focus, which 
 however is not the locus of the intersection of (a) and (b); 
 
122 PLANE ANALYTIC OEOMETRT. 
 
 for although these values of x and y satisfy {b), they do not 
 satisfy {a). We conclude, therefore, that the latter, namely, 
 
 x^Q, (11) 
 
 is the equation of the required locus, which is the tangent at 
 the origin or the axis of Y. 
 
 109. Pkoblem. To find the locus of the point of inter- 
 section of tivo ta?igents to a parabola wJiich are perpendicular 
 to each other. 
 
 Let the equation of one of the tangents he 
 
 2/ = »^ + |^- (§103) {a) 
 
 Then the equation of the other, j^erpendicular to (a), is 
 
 or my = — x — ipni", (b) 
 
 Multiplying (a) by 7n and subtracting (b) gives 
 
 = 2(1 + 'in')x + (1 + m')p, 
 or x= — ip, (12) 
 
 the equation of the required locus, which is the directrix. 
 
 110. Problem. To find the length of the perpendicular 
 from the focus uyon the tangent at any ptoint. 
 
 Let P denote the length of the perpendicular. The equa- 
 tion of the tangent at the point {x', y') is 
 
 y'y-p{x-\-x')^o, (§102) 
 
 The perpendicular P from the focus, whose co-ordinates 
 are (i;;, 0), is (§ 41) 
 
 ^^ p{p-\-'Zx' ) ^ p{2x'-}-p) 
 2Vjr-\-p^ 2V2px' -{-p' 
 = i i^pip + 3a;') 
 = ^V2pr, (§99) (13) 
 
 where r is the focal distance of the point of tangency. 
 
THE PARABOLA. 
 
 123 
 
 111. Problem. To find the co-ordinates of the point of 
 contact of a tangent drawn from a given point to a parabola. 
 
 Let {x', y') be the co-ordinates of the required point of 
 contact, and {h, k) the co-ordinates of the given fixed point. 
 The equation of the tangent at (x', y') is 
 
 and since the tangent passes through the point (h, h), we also 
 have 
 
 ky'=p{x'-\-h), {a) 
 
 Because the point {x', y') is on the curve, we have 
 
 y'- — "Ipx' 
 
 (S) 
 
 Solving (ci) and (^) for x' and y* , we have 
 x' = F - ^h ± h \/Tc 
 
 2ph', 
 
 These equations show that from any fixed point two tan- 
 gents can be drawn to a parabola, and that the points of con- 
 tact {x\ y') will be real, coincident or imaginary according 
 as k"^ — 2/)/i > 0, =0, or < 0; that is, according as the 
 point {h, k) is ivithout, on or icithin the curve. 
 
 11^. Problem. To find the equation of the paralola 
 referred to any diameter and the tangent at its vertex, as axes. 
 
 Let A' be any point on a 
 parabola; take this point as 
 origin and draw through it 
 the diameter A'X' for the 
 new axis of X, and the tangent 
 TA'Y' for the new axis of Y. 
 
 Let Y'A'X'=A'TX=e, 
 and h and k the co-ordinates 
 of A' referred to the original 
 axes AXy AY. 
 
 Let ix, y) be the co-ordi- 
 nates of any point P referred to the original axes, and (re', y') 
 the co-ordinates of the same point referred to the new axes; 
 draw the ordinates PM, PM' , and draw if'iV^and-4'^paral- 
 
124 PLANE ANALYTIC GEOMETRY. 
 
 lei to A Y, and let Q denote the intersection of the diameter 
 A'X' and the ordinate FM. Then, from the figure, we have 
 
 x = AM= AH-i- A'M' + M'Q 
 
 = h^x' -{- PM' cos PM'Q 
 = h -\- x' -{- y' cos dy (a) 
 
 and y = PM=A'H+PQ 
 
 = ]c-\- PM' sin PM'Q 
 
 = k ^ y' sin 0. {b) 
 
 But since the point {x, y) is on the curve, 
 
 y' = 2px. (c) 
 
 Substituting the values of x and y as given by (a) and (b) 
 in (c), we have 
 
 (k 4- 2/' sin ey = 2p(h + x' + y' cos ^); 
 whence 
 
 y" sin' + 22/'(^ sin 6^ -i? cos 6)-^ ¥ - 2jjh = 22)x\ 
 
 But, by (6) of §101, 
 
 Jc =p cot 6; 
 
 and since the point {h, Jc) is on the curve 
 
 F = 2ph, 
 therefore we have 
 
 y'^ sin' = 2px' , 
 
 which is the equation of the curve referred to the new axes. 
 
 Cor. This equation may also be expressed in terms of 
 A'F, the focal distance of the point A\ Thus, by (3) of §99, 
 
 A'F=ip-\-h, 
 
 k* 
 and k^ = 2ph, or h = ^. 
 
 k^ 
 Therefore A'F=ip-\--- 
 
 ^p 
 
 = i{p-\-pooi^e) 
 
 (since k=p cot 6) 
 = ip(l 4- cot' d) 
 
 ^ P 
 2 sin' 6' 
 
THE PARABOLA. 125 
 
 Therefore, denoting A'Fhj ^p', equation (14) may be written 
 
 or, suppressing the accents on the variables, 
 
 y' = 2p'x, (15) 
 
 Cor. From the identity of form in the equations 
 
 ^' = 2px and y"" = )lp'Xy 
 
 we may at once infer that the equation of the tangent referred 
 to any diameter is 
 
 y'y=p'{x-^x'). (16) 
 
 If in this equation we put j^ = 0, we get 
 
 X ^= — X''y 
 
 or, the intercept on the axis of X is equal to the abscissa of 
 the point of contact, and therefore the subtangent to any 
 diameter of a parabola is bisected by the vertex. 
 
 Poles and Polars. 
 
 113. Def. A chord of contact is the chord joining 
 the points of contact of two tangents. 
 
 Problem. To find the equatio7i of the chord of contact 
 of tiuo tangents from an external point. 
 
 Let (ojj, yj be the co-ordinates of the external point, 
 (a;', y') the co-ordinates of the point where one of the tangents 
 through (x^j y^ meets the curve, and {x", y") the co-ordi- 
 nates of the point where the other tangent meets the curve. 
 
 The equation of the tangent at (x' , y') is 
 
 ify=p{x' -^x); ^ (a) 
 
 and since this passes through (x^, ?/j), we have 
 
 and, for the same reason, 
 
 yy = p(x'' -}- x;). (c) 
 
 Hence the equation of the chord of contact is 
 
 y^y =p{x-{-x;), (17) 
 
126 PLANE ANALYTIC GEOMETRY. 
 
 for this is the equation of a straight line, and is satisfied by 
 
 X = x\ y = y' and x = x" , y — y", 
 
 as we see from (h) and (c). Therefore (17) is the equation of 
 the chord of contact of the tangents through the poi^it {x^, y^, 
 
 114, Locus of the Poiiit of Intersection of Two Tangents. 
 
 Let {x^, y^ be the co-ordinates of any fixed point through 
 which a chord of contact of two intersecting tangents is drawn, 
 and (a;^, i/J the co-ordinates of the point of intersection of the 
 taugents. Then the equation of the chord of contact is, by 
 the last section, 
 
 y^y =i?(^\ + ^); 
 
 but since {x^, y^ is a point on the chord, we must also have 
 the condition 
 
 y^y, = PC^\ + ^^)y 
 
 which the co-ordinates of the point of intersection must al- 
 ways satisfy, however the chord of contact may change its 
 position as it revolves about the fixed point (x^^ y^. Therefore 
 the equation of the required locus is 
 
 y^y = p{^ + ^^)y (IS) 
 
 which is that of a straight line. Hence we have the theorem: 
 
 If through any fixed point tve draw chords to a parabola; 
 
 and if through the ends of each chord we draiu a pair of 
 tangents, 
 
 then the point of meeting of every pair of tangents will lie 
 on a certain straight line. 
 
 Def Such straight line is called the polar of the point 
 through which the chords pass. 
 
 It follows from this theorem that if (x^, y^) be any fixed 
 point, the equation of the polar of that point is 
 
 y,y = p)(x + x^) (19) 
 
 when referred to the axis, or 
 
 y.y = p\^ + ^.) (^0) 
 
 if referred to a diameter and a tangent at its vertex as axes. 
 
THE PARABOLA. 127 
 
 Direction of the Polar. 
 Making y^ = in (20), we have 
 
 x= ~ a;,, (21) 
 
 which is the equation of a line parallel to the axis of Y. Hence 
 The polar of any point is parallel to the tangent at the end of 
 
 the diameter ptcLssing through that pointy and is situated at a 
 
 distance from the vertex of the diameter equal, but in an optpo- 
 
 site direction, to the distance of the point. 
 115. Polar of tlie Focus. 
 Put {^p, 0) for x^, y^ in the equation of the polar, and we 
 
 get 
 
 x= - ip, (22) 
 
 which is the equation of the directrix. Therefore 
 Tlie polar of the focus of a parabola is the directrix. 
 
 EXERCISES. 
 
 1. Find the points of intersection of the line ?/ = 3a; — 6 
 with the parabola y"^ — 9:c. Ans. (4, 6) and (1,— 3). 
 
 2. Find the equation of a line through the focus of the 
 
 parabola ?/^ = 12rc and making an angle of 30° with the axis 
 
 of iC. . X ^ 
 
 ^t^8. y = —-^, 
 
 3. Find the equation of the line through the vertex and 
 the extremity of the latus rectum. Ans. y = ± ^x. 
 
 4. Find the equation of the circle which passes through 
 the vertex of a parabola and the extremities of the latus rec- 
 tum. Ans. x^ -\- y^ = ^px. 
 
 5. Find the equation of the tangent at the extremity of 
 the latus rectum, and the angle between this tangent and 
 the line drawn to the vertex from the same extremity of the 
 latus rectum. Ans. y = x -{- ^p; tan^~^^i. 
 
 6. Determine the equations of the normals at the extrem- 
 ities of the latus rectum, the co-ordinates of the points in 
 which these normals again intersect the curve, and the length 
 of the chords formed by the normals. 
 
128 PLANE ANALYTIC GEOMETRY. 
 
 7. Show that if the focus of a parabola is the origin, and 
 the axis of the curve the axis of X, the equation of the para- 
 bola is y^ = p(2x + jy), and the equation of the tangent at 
 the point {x\ y') is 
 
 y'y = p{^ + ^' +i?). 
 
 8. With the same origin and axes as in the last example 
 show that the equations of the tangents and normals at the 
 extremity of the latus rectum are 
 
 X T y -\- p = 0; 
 X ± y — p = 0, 
 
 9. Prove that the circle described on any focal chord as 
 diameter will touch the directrix. 
 
 10. A tangent is drawn to a parabola at the point (.^•', ?/'). 
 Find the length of the perpendicular drawn from the foot of 
 the directrix on this tangent. 
 
 Ans. ^ ^ 
 
 )lVy'^-\-p^ 
 
 11. Pairs of tangents are drawn to a parabola at points 
 whose abscissas are in a constant ratio. Show that the locus 
 of the intersection of the tangents is a parabola. 
 
 12. Find the polar equation of the parabola when the 
 vertex is the pole, and the axis of the curve the initial line. 
 
 Alls, r = 2p cot 6 cosec 6. 
 
 13. If r and r' be the lengths of two radii vectores drawn 
 at right angles to each other from the vertex of a parabola, 
 show that 
 
 14. Find the equation of the parabola referred to the tan- 
 gents at the extremities of the latus rectum as axes. 
 
 Ans. {x — yY — 2 V2p{x + y) -^2p' = 0. 
 
 15. If tangents be drawn to a parabola at the extremities 
 of any focal chord, show that they will intersect at right 
 angles on the directrix, and that the line from their point of 
 intersection to the focus is perpendicular to the focal chord. 
 
 'Jr/r- 
 
THE PARABOLA. 129 
 
 16. From an external point {x\ if) two tangents are drawn 
 to a parabola. Show that the length of the chord of contact is 
 
 
 
 2(ir 
 
 J^f)\y'^-2px')^ 
 
 
 
 P 
 
 3 
 
 and that the 
 
 area 
 
 of the 
 
 J triangle formed 
 
 by the chord and tan- 
 
 gents is 
 
 
 
 P 
 
 
 17. If ni, m' be the slopes to the axis of the parabola of 
 the two tangents in the last example, show that 
 
 m-\-m' = ~ and mm' = r^. 
 
 X n/X 
 
 18. If {x', y') and (re", y") be any two points on a para- 
 bola, show that the tangent of the angle contained by the 
 tangents touching at these points is 
 
 p{y" - y') 
 f + y"y' 
 
 19. In what ratio does the focus of a parabola divide that 
 segment of the axis cut out by a tangent and normal drawn 
 at the same point of the parabola? 
 
 20. A triangle is formed by three tangents to a parabola. 
 Show that the circle which circumscribes this triangle passes 
 through the focus. 
 
 21. Show that the parameter of any diameter is equal to 
 four times the focal distance of its vertex, or equal to the focal 
 double ordinate of that diameter. 
 
 Note. The parameter of any diameter is the focal chord bisected 
 
 by that diameter, called 2y in § 112. 
 
 I 
 
 22. If TP and TQ are tangents to a parabola at the points 
 P and Qi then if F be the focus, show that 
 
 FP . FQ= FT\ 
 
130 PLANE ANALYTIC OEOMETRT. 
 
 23. Sliow that tlie area of the triangle in Prob. 20 is half 
 that of the triangle formed by joining the points of contact 
 of the three tangents. 
 
 24. Given the outline of a parabola, show how to find the 
 focus and the axis. 
 
 25. The base of a triangle is 2a, and the sum of the tan- 
 gents of the base-angles is m. Show that the locus of the ver- 
 tex is a parabola whose semi-parameter is — . 
 
 26. Prove that y — x tan +jt? cosec 20 is a tangent to 
 a parabola whose latus rectum is p, the origin being at the 
 focus, and the axis of the curve the axis of X. 
 
 27. Tangents are drawn from any two points P, § to a 
 parabola. Show that the co-ordinates of T, the intersection 
 of the tangents, are 
 
 1 cos {B, + e,) \ sin ((9, -f (9,) 
 4^ sin 0, sin 0/ 4^^ sin 0, sin 0/ 
 
 where tan Q^ and tan 6^ are the slopes of the tangents to the 
 axis of X, 
 
 28. If all the ordinates of a parabola are increased in the 
 same ratio, show that tlie new curve will be a parabola, and 
 express its parameter in terms of the ratio of increase. 
 
 29. At what point of a parabola is the normal double the 
 subtangent; and what angle does that normal form with the 
 axis of the parabola? 
 
 30. Find a point upon a parabola such that the rectangle 
 contained by the tangent and normal shall be twice the square 
 of the ordinate; and show the relation of such point to the 
 focus. 
 
 31. Find that point on a parabola for which the normal is 
 equal to the difference between the subtangent and the sub- 
 normal. 
 
 32. Having given the parabola if — 6a', find the equation 
 of that chord wliich is bisected by the point (4, 3). 
 
 33. Find the equation of that chord of a parabola which 
 is drawn from the vertex and bisected by the diameter y — q* 
 
CHAPTER VI. 
 TH E ELLI PSE 
 
 Equations and Fundamental Properties. 
 
 116. Def. An ellipse is the locus of a point the sum 
 of whose distances from two fixed points is constant. 
 
 The two fixed points are called foci of the ellipse. Thus, 
 if the point P move in such a way that PF -\- PF' is con- 
 stant, it will describe an ellipse. 
 
 The curve may be described me- 
 chanically as follows: Take any two 
 fixed points i^'and F' , and attach to 
 tbera tbe extremities of a thread whose £ 
 length is greater than the distance FF' . 
 Place a pencil-point P against the 
 thread, and slide it so as to keep the 
 thread constantly stretched: the point 
 P will describe an ellipse, for in every 
 position we shall have PF -\- PF' = the constant length of the thread. 
 
 The line AA^ drawn through the foci and terminated by 
 the curve is called the transverse or major axis, and BB' 
 bisecting AA' at right angles is called the conjugate or 
 minor axis. The two are called principal axes. 
 
 The semi-axes CA and CB are represented by the symbols 
 a and i respectively. 
 
 The point C midway between 
 the foci is called the centre. 
 
 From the manner in which the 
 curve is generated, we see that A I 
 
 AF= A'F' 
 and 
 
 PF-\. PF' = AA'. 
 
132 PLANE ANALYTIC GEOMETRY. 
 
 117. Problem. To find the equation of the ellipse. 
 
 Solution. Let C, the intersection of. AA^ and BB^, be 
 the origin; CA the axis of X, and GB the axis of Y; put 
 CA = CA' = a, CF — OF' = c, and x, y the co-ordinates 
 of any point P on the locus. Then we shall have 
 
 PF 1= VPM' + MF' = Vif + (c - xY; 
 PF' = VPM' + MF'' = Vy' + (c + x)\ 
 Therefore, by definition. 
 
 Clearing this equation of surds, it reduces to 
 
 («' - c')x' + ay = a\a' - c'). 
 But, by definition, 
 
 a' -c' = BF' - CF' = BC = b'; 
 therefore we have, by substituting in the above, 
 
 i^x' + a'y' = a-F; 
 or, dividing by a'^b', we have 
 
 X' . y 
 
 a' 
 
 + f. = 1, (1) 
 
 which is the simplest form of the equation of the ellipse. It 
 is called the equation of the ellipse referred to its centre and 
 axes, because the centre is the origin and the axes are the 
 axes of co-ordinates. 
 
 Def. The distance CF = CF' = c between the centre 
 and either focus is the linear eccentricity of the ellipse. 
 
 The ratio — of the linear eccentricity to the semi-major 
 
 axis is called the eccentricity of the ellipse. 
 By the common notation, 
 
 a a ^ ' 
 
 is the expression for the eccentricity in terms of the semi- 
 axes. 
 
THE ELLIPSE. 133 
 
 Oor, If we transfer the origin to A', whose co-ordinates 
 are (— a, 0), the equation (1) becomes, by writing {x — a) 
 for Xy 
 
 (^ - ctY , .1' _ 1 
 
 or y'"^a' ^^^^ ~ ''^')' 
 
 a form of the equation of the ellipse which is sometimes use- 
 ful. 
 
 EXERCISES. 
 
 1. Find the eccentricity and semi-axes of the ellipse 
 
 16x' + 26 f = 400. 
 
 Remark. Reduce the second member to unity by dividing by 400, 
 and compare with (1). 
 
 2. What are the semi-axes and the equation of the ellipse 
 when the distance between the foci is 2 and the sum of the 
 distances from each point of the curve to the foci is 4? 
 
 3. Determine the eccentricity and semi-axes of the ellipses 
 having the following equations: 
 
 (a) x' + 2y' = 6; (b) 3x' + 4/ = 9; (c) 4a:' + 9i/ = 16; 
 (d) mx' + nf =p; (e) ^x^ + 1^ = ^ if) «^' + if = L 
 
 4. Using the preceding notation, prove the following pro- 
 positions: 
 
 I. The distance of either focus from the centre is ae. 
 
 II. The distance of either focus from the nearest end of 
 the major axis is a{l — e). 
 
 III. The distance of either focus from the farther end of 
 the major axis is a(l -f- e). 
 
 IV. The distance from either end of the major axis to 
 either end of the minor axis is a V2 — e^. 
 
 V. If we define an angle cp by the equation 
 
 sin q) = e, 
 we shall have for the semi-minor axis 
 h = a cos q}. 
 
134 
 
 PLANE ANALYTIC OEOMETRY. 
 
 5. Find the points in which the circle x" -\- y^ = ^ inter- 
 sects the ellipse x"" -f ^if — G. 
 
 6. Write the equation of that ellipse whose minor axis is 
 10 and the distance between whose foci is 12. 
 
 118. If we solve equation (1) with respect to y, we find 
 
 h 
 
 y 
 
 ± - Va' - x\ 
 a 
 
 This equation shows that for every value of x there will 
 be two values of y, equal but with opposite signs. Hence the 
 curve is symmetrical with respect to the major axis. 
 
 By solving with respect to x we show in like manner that 
 the curve is symmetrical tvith respect to the 7ninor axis. 
 
 Def. A chord of an ellipse is any straight line terminated 
 by two points of the ellipse. 
 
 A diameter of an ellipse is any chord through its centre. 
 
 Cor. The major and minor axes are diameters. 
 
 Def. The parameter or latus rectum is a chord 
 through the focus and perpendicular to the major axis. 
 
 119. Theorem I. The parameter of an ellipse is a third 
 proportiojial to the major and minor axes. 
 
 Proof. The semi-parameter is, by definition, the value of 
 the ordinate y when x = ae. From equation (1), we have 
 
 
 x^). 
 
 tion in this equation. 
 
 P 
 
 Hence p 
 
 a- 
 
 ~ a 
 a : l 
 
 («» 
 
 If we put p for the semi-parameter, we find, by substitu 
 
 h^ 
 a'' 
 
 or a : = h : p. 
 
 Cor. The length of the semi-parameter FL is 
 p = a{l- e'). 
 
 or 
 
 ap = 
 
 (3) 
 
TEE ELLIPSE. 
 
 135 
 
 120. Focal Radii, or Radii Vedores. 
 
 Def. The focal radii of an ellipse are the lines drawn 
 
 from any point on the curve to the foci. 
 
 Problem. To express the lengths 
 of the focal radii in terms of the ab- 
 scissa of the point from which they 
 are draion. 
 
 Let r and r' be the focal radii of 
 the point P, whose co-ordinates are 
 
 {^^ y)' 
 
 Becanse FG = OF' = ae, 
 we have r' = FM' + PM' 
 = (x — ae)' + y^ 
 
 = {X - aey + ^, {a^ - x^) 
 
 = x' - 2aex + a'e' + (1 - e'){a' - x") 
 = a^ — 2aex -j- e'^x^. 
 Therefore r = a — ex. (4) 
 
 In the same way we find, for the other focal radius, 
 
 7*' = rt + ex. (5) 
 
 These expressions are of remarkable simplicity, and, being 
 of one dimension in x, either of them is called the linear 
 equation of the ellipse. 
 
 We observe that their sum is 2a, as it should be. 
 
 Cor. Equations (4) and (5) show that if a point move on 
 the circumference of an ellipse in such a way that its abscissa 
 increases unifoi^mly, one focal radius tvill increase and the 
 other will decrease uniformly. 
 
 In other words, if the abscisses of several points are in 
 arithmetical jjrogi'ession, their focal radii ivill also be in 
 arithmetical progression. 
 
 121. Polar Equation of the 
 Ellipse, the right-hand focus being 
 the pole. 
 
 Let r and 6 be the polar co- 
 ordinates of any point P on an 
 ellipse; that is, r = FP and 
 
136 PLANE ANALYTIC GEOMETRY. 
 
 6 = the angle AFP. Join PF'. Then, from the triangle 
 FPF', we have 
 
 PF"" = PF' 4- FF" - 2PF. FF . cos PFF, 
 But FF' = 2ac and cos PFF' = - cos AFP; 
 
 therefore PF' = Vr' + Aa'e' + 4.aer cos 6, 
 and by the fundamental property of the ellipse we have 
 PF-\- PF' = A A', 
 
 or r + V?-"" + 4a'e^ -f 4aer cos 6 = 2a; 
 
 whence we easily find 
 
 which is the required equation. 
 
 The polar equation may also be easily obtained from the 
 linear equation of the ellipse; thus, from (4), we have 
 
 r = a — ex, 
 
 the origin being at the centre. 
 
 Transferring the origin to the right-hand focus, whose co- 
 ordinates are (ae, 0), it becomes 
 
 r = a(l — e"^) — ex, 
 
 which in polar co-ordinates becomes 
 
 r = a(l — e"^) — er cos 6; 
 
 whence r = zr-^. ^-, 
 
 1 + e cos t^ 
 
 as before. 
 
 If the left-hand focus be taken as the pole, the student 
 may easily show thiit the polar equation is 
 
 a(l - e -") 
 1 — e cos 6' 
 
 Cor. It = 0, we have r = ^\ . ^ - = a(l — e), which 
 is the value of AF. 
 
THE ELLIPSE. 137 
 
 When 6 = 180°, we get r = a(l + e), which is the value 
 of A'F. 
 
 When 6 = 90°, r = a(l — e^), the semi-parameter. 
 These results agree with those of §§ 117, 119. 
 
 EXERCISES. 
 
 1. If the semi-minor axis of an ellipse is b, and the eccen- 
 tricity sin cpj express its semi-major axis and semi-parameter 
 in terms of b and cp. Ans. a^=h sec q)\ 
 
 p z= h cos <p. 
 
 2. The distance from the focus to the nearer end of the 
 major axis is 2, and the semi-parameter is 3. Find the 
 major and minor axes and the eccentricity. 
 
 3. Express the ratio of the parameter to the distance be- 
 tween the focus and either end of the major axis. 
 
 4. The major axis is divided by the focus into two seg- 
 ments. Show that the rectangle contained by these segments 
 is equal to the rectangle contained by the semi-major axis and 
 the semi-parameter, and also equal to the square of the semi- 
 minor axis. 
 
 5. Write the equation of an ellipse in terms of its semi- 
 minor axis h, and its semi- parameter j(?. 
 
 Ans. fx^ 4- yy'' = ^"' 
 
 6. Find the points in which the several straight lines 
 
 y = %x, y^lx^l, y = 2x + 2, 
 
 intersect the ellipse x'^ + 2y^ = 6, and the lengths of the three 
 chords which the ellipse cuts out from the lines. 
 
 7. Find the equation of the ellipse when the right-hand 
 focus is the origin, the axes being the major axis and the 
 latus rectum. 
 
 x'^ , w"* , 2ex F 
 Ans. -o + fo H = -o. 
 
 8. The sum of the principal axes of an ellipse is 108, and 
 the linear eccentricity 3G. Find the equation of the ellipse, 
 and the eccentricity. 
 
 Ans. 39". + 15^ = 1; ^ "^ 13* 
 
138 PLANE ANALYTIC OEOMETRT. 
 
 Diameters of an Ellipse. 
 
 122. Theorem II. Every diameter of an ellipse u hi- 
 sected hy the centre. 
 
 Proof. Let y = mxhe the equation of any line through 
 the centre. Eliminating y between this equation and that of 
 the ellipse, we have 
 
 a''^~¥~~ 
 
 from which to determine the abscissae of the points in which 
 the line intersects the ellipse. Since this equation contains 
 terms in x^ but none in x, it will reduce to a pure quadratic, 
 of which the two roots are equal but with opposite signs. 
 From these roots we shall get, by substituting in the equation 
 y = mx, two equal values of y with opposite signs. Hence 
 the points of intersection are at equal distances on each side 
 of the origin. 
 
 123. Theorem III. The locus of the centres of parallel 
 chords of an ellipse is a diameter. 
 
 Proof. Let y = mx -{- h (a) 
 be the equation of a chord; the 
 slope m being the same for all 
 the chords, while h varies from 
 one chord to another. 
 
 We first find the points of in- 
 tersection of the chord with the ellipse in the usual way. 
 
 Eliminating 2/ between (a) and the equation of the ellipse, 
 we find the abscissae of the points of intersection to be deter- 
 mined by the quadratic equation 
 
 x^ (mx -f hY _ 
 
 which being reduced to the general form becomes 
 
 _2aM_ a\h'- b ') _ 
 "" + a'm' -f Z^^^ "^ a'm'' + I' 
 
 Now we need not actually solve this equation to obtain 
 
THE ELLIPSE. 139 
 
 the result we want, namely, the abscissa of the middle point 
 of the chord. We know that if we put, for brevity, 
 
 ^ ~ a'm' + b' ' 
 and call the roots x^ and x^, we shall have 
 
 which give the abscissas of the two points in which the chord 
 intersects the ellipse. The corresponding values of y, from 
 
 (a), are 
 
 y^ z=z mx^ + h; 
 
 y^ = mx, + h. 
 
 By (§23), the co-ordinates of the middle point of the 
 chord are the half -sums of the co-ordinates of the extremities. 
 If, then, we put x\ y' for the co-ordinates of the middle point 
 of the chord, we have 
 
 , _ ^ _ ct^mh 
 
 y' = im{x^ + x^) + h 
 = rnx' -\- h 
 
 bVi 
 
 The problem now is, What relation exists between x' and 
 y' when we suppose h to vary and all the other quantities which 
 enter the second member of (b) to remain constant? We ob- 
 tain this relation by eliminating h between the two equations, 
 which is done by multiplying the first by b^ and the second 
 by a'm and adding the products. Thus we find 
 
 b'x' + a'm/^O. ^ (7) 
 
 This is a relation between the co-ordinates of the middle 
 points of the parallel chords which is true for all values of h, 
 
140 PLANE ANALYTIC GEOMETRY. 
 
 that is, for nil such chords; it is therefore the equation of 
 the required locus, and, from its form, is a straight line 
 through the origin and therefore through the centre of tlie 
 ellipse. 
 
 124. Conjugate Diameters. If we omit the accents in 
 (7), we may write it in the form 
 
 ^ am 
 
 By assigning different values to m, or, which is the same 
 thing, by giving different directions to the parallel chords, the 
 
 slope ^- may take all possible values, and therefore (7) 
 
 may represent any line passing through the centre and bisect- 
 ing a system of parallel chords. 
 
 If m' be the slope of the diameter which bisects all the 
 chords whose slope is m, we have 
 
 y z= m'x, 
 
 the equation of the diameter; 
 
 but, by (7), y = =— x 
 
 •^ ^ ^ ^ a^m 
 
 is also the equation of the diameter. 
 
 Therefore m* = 5--, 
 
 am 
 
 or mm' = 5-. (8) 
 
 a ^ ' 
 
 Theorem IV. If one diameter bisects chords parallel to 
 a second diameter, the second dia^neter will bisect all chords 
 parallel to the first. 
 
 Proof. If m and m' be the respective slopes of the two 
 diameters, we shall have 
 
 ^' 
 
 mm = 5-, 
 
 a 
 
 since the first bisects all chords parallel to the second; but 
 this is also the only condition which must hold in order that 
 the second may bisect all chords parallel to the first. 
 
THE ELLIPSE. 
 
 141 
 
 Def. Two diameters each of which bisects all chords par- 
 allel to the other are called conjugate diameters. 
 
 Cor. As the chords of a set become indefinitely short near 
 the terminus of the bisecting diameter, they coincide in direc- 
 tion with the tangent at the terminus. Hence: 
 
 Theorem V. The tangent to an ellipse at the end of a 
 diameter is parallel to the conjugate diameter. 
 
 125, Problem. Given the co-ordinates of the extremity 
 of one diameter, to find those of either extremity of the con- 
 jugate diameter. 
 
 Solution. Let P CP' and D CD' p^-- --^ p 
 
 be any pair of conjugate diameters, 
 and (x', y') the given co-ordinates 
 of P. Then the equation of GP is 
 
 
 y 
 
 -^Lx 
 
 (since m 
 
 -yL\ 
 
 ~ xn 
 
 and the 
 
 equation 
 
 of DD' is 
 
 
 
 
 
 y = - 
 
 am 
 
 or 
 
 
 
 y^- 
 
 ¥x' 
 
 and the equation of the ellipse, 
 
 aY + 2''^' = a'^'. 
 Substituting from {b) in (c), we have 
 
 {.Vx'-" + a'y''')x^ = aY'; 
 but since («', ?/') is on the ellipse, we have 
 b^x'' + a'y'' = a'b'; 
 therefore a^'b^x'' = a*y'^y 
 
 (x,y) 
 
 (0) 
 
 or 
 
 X=±p' 
 
 Substituting this value of x in (a), we get 
 
 ^ a 
 
142 
 
 PLANE ANALYTIC QEOMETRT. 
 
 136. Theorem VI. The sum of the squares of tioo con- 
 jugate semi-diametei^s is constant and equal to the sum of the 
 squares of the semi-axes. 
 
 Proof Let (x', «/') be the co-ordinates of P (last figure), 
 and denote the semi-conjugate axes CP, CD by a' and b' 
 respectively. Then we shall have 
 
 CP» + CD^ = x'^ + 2/" + ^y-- + l^;- 
 
 _a'b' aW 
 ~ b' '^ a' ' 
 
 or 
 
 a" + b" = a'-\- b\ 
 
 (9) 
 
 12*7. Problem. To find the angle between two cofijugate 
 axes. 
 
 Let 6 and 6' be the angles which ^ 
 the semi-conjugate axes make with ' "^ 
 the major axis, and cp the angle be- 
 tween the conjugate axes. Then 
 
 and 
 
 (p = e' -e 
 
 sin cp = sin 6' cos — sin ^ cos 6 
 
 Denote the semi-conjugate axes by a' and b', and the co- 
 ordinates of P by x', y'. Then (§ 125) the co-ordinates of D 
 are 
 
 -p'' 
 
 +^'• 
 
 Hence sin Q = ^-; 
 
 cos u = — ,-; 
 
 • af ^^' 
 
 cos^'= ,,- 
 
 Substituting in (a), we have 
 
 
 bx" , 
 ^'" ^ = aa'b' + 
 
 
THE ELLIPSE. 
 
 143 
 
 (10) 
 
 But since (?/', y') is on the curve, 
 
 Therefore sm a> = , ,,, = -nr- 
 
 128. Theorem VII. The area of the parallelogram which 
 touches an ellipse at the ends of conjugate diameters is constant 
 and equal to the area of the rectangle ivhich touches the elli2)se 
 at the ends of the axes. 
 
 Proof From the last equation we have a'b' sin (p = ab; 
 but a'b' sin cp is equal to the area of the parallelogram CPQD, 
 and ab is equal to the area of the rectangle CAEB; therefore 
 the parallelogram QRST = the rectangle EFGH, which is 
 constant. 
 
 Cor. 1. The triangle CPD is equal to the triangle ACB, 
 each being one half of the parallelograms QC and. ^(7 respec- 
 tively. 
 
 Cor. 2. If P denote the perpendicular from C on QT, we 
 have 
 
 P . CD = area of CPQD 
 = ab. 
 
 Therefore P' = ^^ = jr-r' 
 
 But, by § 126, b'' = a' + b' - a'\ 
 
 21,3 
 
 Hence P' =f , , "!; ti- (H) 
 
144 
 
 PLANE ANALYTIC OEOMETRT. 
 
 129. Problem. To find the equation of the ellipse re- 
 ferred to a pair of conjugate diameters as axes. 
 
 Let CP, CD be any two conju- 
 gate semi-diameters; take CP for 
 the new axis of X, and CD for the 
 new axis of Y', let the angle 
 ACP=a, and the angle A CD =13', 
 {x, y) the co-ordinates of any point 
 Q of the ellipse referred to rect- 
 angular axes, and {x', y') the co-ordinates of the same point 
 referred to the new axes. 
 
 The formulae for passing from rectangular to oblique axes 
 are (§ 29) 
 
 a; = a;' cos a: + y' cos /?; 
 y = x' sin a -\- y' sin /?. 
 
 But since {x, y) is on the ellipse, we have 
 
 a'y' + h'x' = a'b\ 
 Eliminating x and y from these three equations, we have, 
 after reduction, 
 
 (a" sin'a + h^ cosV)a;" -f (a« sin'y5 + h' cos' /3)y'' 
 
 + 2{a' sin asm/3 -\- V cos a cos §)x'y' = c^W. 
 But since CP and CD are conjugate semi-diameters, we have, 
 by (8), the condition 
 
 mm' 
 
 or 
 
 that is, 
 
 tan a tan p = 
 sin a sin /3 
 
 a'' 
 a'' 
 
 .2 > 
 
 cos a cos ^ a' 
 
 or «' sin o' sin y^ + ^^ cos a cos ^ = 0, 
 
 Therefore the coefficient of x'y' vanishes and we have 
 
 (rt' sin^a + Z>'cosV)a^"+ (a" sin'/? -f J' cos' /3)y''= a'b\ (12) 
 which is the equation of the ellipse referred to the new axes. 
 By putting y'' = 0, we get 
 
 a'F 
 
 a' sin'a -j- b' cos'a 
 
 = CP\ 
 
THE ELLIPSE. 
 
 145 
 
 which we have already denoted by «". In a smiihir manner 
 we get 
 
 which we have denoted by b'"^. 
 Hence (12) may be written 
 
 — 4-^-1- 
 
 or, suppressing the accents on the variables, since the equation 
 is entirely general. 
 
 -1- J/. 
 
 (13) 
 
 Comparing this with (1), we see that the equation of the 
 curve referred to the major and minor axes is only a particu- 
 lar form of the more general one which we have just obtained. 
 From the identity of form in (1) and (13) we see that the 
 transformations of the former are applicable to the latter; 
 therefore it follows that any formulae derived from the equa- 
 tion of the ellipse by processes which do not presuppose the 
 axes to be rectangular will be applicable when any pair of 
 conjugate semi-diameters are substituted for the principal 
 semi-axes. 
 
 130. Supplemental Chords. 
 
 Def. The two straight lines drawn from any point on an 
 ellipse to the extremities of any diameter are called supple- 
 mental chords. 
 
 If the diameter is the major axis, the chords are called 
 principal supplemental chords. 
 
 Relation between Two Supplemen- 
 tal Chords. Let PP' be any diame- 
 ter, andP§, P'Q two supplemental 
 chords; {%', y') the co-ordinates of 
 P, and therefore (— x', — y') the 
 co-ordinates of P', and {x, y) the. 
 co-ordinates of Q. Then the equa- 
 tion of the line PQ may be written 
 
 y - y' ^m{x - x'), 
 
146 
 
 PLANE ANALYTIC GEOMETRY. 
 
 iind the equation of the line P' Q may be written 
 
 y -\- y' = fn\x + x')', 
 
 whence, by multiplication, 
 
 y' - y"^ mm'[x' - x"), (a) 
 
 But since the points {x, y) and {x' , y') are on the curve, 
 we have 
 
 a'y^ + h'x' = a'b' 
 and ay + Fx"=z a'b'; 
 
 whence a'(?/' - y'') + b^x' - x'') = 0, 
 
 or 
 
 (i) 
 
 Comparing (a) and (b), we have 
 
 b' 
 mm' = U-, 
 
 which is the condition that holds for conjugate diameters 
 whose slopes to the major axis are 7n and m' respectively 
 (§124); therefore— 
 
 Theorem VIII. If any chord of an ellipse is parallel to a 
 diameter, the supplemental chord is parallel to the conjugate 
 difljneter. 
 
 Relation of the Ellipse and Circle. 
 
 131. Let a circle be described on the major axis of an 
 ellipse as a diameter; its equation 
 referred to the centre as origin is 
 
 yc' = a" - x\ (a) 
 
 where y^ represents the ordinate 
 F'M, 
 
 The equation of the ellipse gives 
 
 ye = ^(a^ 
 
 x^), (b) 
 
 Comparing (a) and (b), we have 
 
 y? = ^^ "' 
 b 
 
 ye 
 
 whence 
 
 a 
 
TEE ELLIPSE. 
 
 147 
 
 that is, the ordinate of the ellipse at any point is found by 
 multiplying the ordinate of the circle by the constant factor 
 
 -. Hence we have 
 a 
 
 Theorem IX. If all the ordinates of a circle he dimin- 
 ished in the same proportion^ the circle zuill be changed into 
 a?i ellipse. 
 
 133. The Eccentric Angle. 
 
 Def. If we join P and C, the centre of the ellipse, the 
 angle P'CA is called the eccentric angle of the point P. 
 
 Problem. To express the co-ordinates of any point of the 
 ellipse in terms of the eccentric angle of that point. 
 
 Let the eccentric angle = 9, and x, y, the co-ordinates 
 of the point P. Then, since P^G = AG, we shall have 
 
 X = a cos q)', 
 y = - P'M 
 
 = — a sin (z? = § sin w. 
 
 a ^ ^ 
 
 133. Problem. To find the area of an ellipse. 
 
 Describe a circle on the major 
 axis as a diameter, which we can con- 
 ceive to be divided into any num- 
 ber of equal parts. At any two ad- 
 jacent points, as M, N, draw the 
 common ordinates MP', NQ', and 
 through P and P' draw PH, P'H' 
 parallel to the axis. Let the ordinates 
 PM, P'M be denoted by y^, and y^ 
 respectively. Then, since the rectangles MH, MR' have the 
 same breadth, namely, MN, they are to each other as their 
 heights MP, MP'-, that is, 
 
 MH_ 
 MH' 
 
 (§131) 
 
 In the same way it may be shown that any other pair of 
 similar rectangles in the ellipse and circle have the ratio of 
 
148 
 
 PLANE ANALYTIC GEOMETRY. 
 
 1) : a, and therefore the sum of all the rectangles in the ellipse 
 is to the sum of all the corresponding rectangles in the circle 
 Sisb : a. 
 
 Now if the number of equal parts into which the axis is 
 divided be increased indefinitely, the sum of all the rectangles 
 in the ellipse will approach the area of the semi-ellipse as a 
 limit, and the sum of all the rectangles in the circle will ap- 
 proach the area of the semi-circle as a limit. 
 
 Therefore we shall ultimately have 
 
 Area of the ellipse _ b 
 Area of the circle "" a 
 
 But the area of the circle = Tra'; therefore we shall have 
 the area of the ellipse = Ttab. Hence: 
 
 Theorem X. The area of an ellipse is a mean propor- 
 tional between the areas of the circles described on the major 
 and minor axes. 
 
 Tangents and Normals to an Ellipse. 
 
 134. Problem. To find the equation of the tangent to 
 an ellipse at a given point. 
 
 Let x', if be the co-ordinates 
 of any point on the curve, and 
 x" , y" the co-ordinates of an 
 adjacent point on the curve. 
 The equation of the secant pass- 
 ing tlirough the points x' , y' 
 and x", y" is, by § 45, 
 
 V 
 
 -,{x - X'), 
 
 ^ ^ X" — X' 
 
 Since {x', y') and {x", y") are on the ellipse, we have 
 ^^y" -I- ^^c" = a'b' 
 
 0: 
 
 (^0 
 
 and 
 
 a'y'^' + ^V" = ct'b'', 
 
 therefore 
 
 «'(/" - y'") + h\^"" - ^'") = 
 
 whence 
 
 y" - ?/' b' x." + x' 
 -jo" - x' " «' • ll" + ?/' 
 
THE ELLIPSE. 149 
 
 Substituting in (a), we have, for the equation of the secant, 
 
 y - y - -^ -fr^i^ - ^')- i») 
 
 Now if the points (a;', y') and {x'\ ?/") approach each 
 other until they coincide, the secant SS^ will become the tan- 
 gent TT\ We shall then have at the limit 
 
 re" = x' and ^" = ?/'; 
 
 hence (b) becomes 
 
 / ox. -. 
 
 y - y = - -.- ^(^ - ^), 
 
 a y 
 
 which is the equation of the tangent at the point x', y'. 
 
 This equation may be simplified thus: Multiply by a^y^ 
 and we get 
 
 a'yy' + h'xx' = a'y'' + b'x'^ 
 
 or %J^yl. = \, (15) 
 
 a 
 
 The equation of the tangent may also be expressed inde- 
 pendently of the coordinates of the point of contact, as fol- 
 lows: 
 
 135. Problem. To find the condition that the line 
 y = mx -f- h 
 may be tangent to the ellipse 
 
 a' ^ b' 
 If we eliminate y between these equations, we have 
 x^ {mx -^ hy _ 
 
 or (Z*' + a''m')x'' + 'Za'mhx = a\b'' - h'), {a) 
 
 for determining the abscissae of the points in which the line 
 intersects the ellipse. Since the line is to be a tangent to the 
 ellipse, the two values of the abscissa will be equal. Now the 
 
150 
 
 PLANE ANALYTIC OEOMETRT. 
 
 condition that this equation may have equal roots is, by the 
 theory of quadratic equations (§ 8), 
 
 whence h' = h' + ce^^\ (16) 
 
 or h = ± Vb' + ci'm', 
 
 the required condition. 
 
 Substituting this value of h m the given equation of the 
 
 line, we have 
 
 y = mx ± Vb' + a'w' (17) 
 
 for the equation of the tangent. 
 
 Conversely, every equation of this form is the equation of 
 some tangent to the ellipse. The double sign shows that there 
 will always be two tangents having a given slope. 
 
 Eemark. From the facility with which this equation en- 
 ables us to solve many problems involving the use of the equa- 
 tion of the tangent, it is sometimes called the magical equa- 
 tion of the tangent. 
 
 136. The 8ubtangent. 
 
 Def. The projection on the axis of Xof that portion of 
 the tangent intercepted between the point of contact and the 
 axis of X is called the subtangent. 
 
 To find where the tangent intersects the axis of X, we 
 make i/ = in the equation of 
 the tangent. Thus the equa- 
 
 tion of the tangent is 
 
 
 ~^N. 
 
 a^'^ b' - ^' 1 
 
 
 /iY\ 
 
 Making ?/ = 0, we have I 
 
 C 
 
 '^ ii j 
 
 . = J=C.. ^ 
 
 
 ___^ 
 
 Subtracting CM or x\ we have 
 
 
 Subtangent = MT 
 _a' 
 
 J _a'-x" 
 
 Cor. The subtangent is independent of b', hence all 
 
THE ELLIPSE. 161 
 
 ellipses described on a common major axis have a common 
 suhtangent for any given abscissa of thepoi?its of coiitact. 
 
 This property enables us to draw a tangent to an ellipse 
 from any point on the curve. 
 
 Thus, let P be any point on the curve; describe a circle on 
 ^^' as a diameter, and produce the ordinate PM to meet 
 the circle in Q. Then if x' is the abscissa CM, we have 
 
 Subtangent of ellipse = 7 — = subtangent of circle = MT, 
 
 X 
 
 Hence if QThe drawn tangent to the circle and meeting A A' 
 produced in T, then, by what has just been proved, Twill be 
 the foot of tangent to the ellipse at P, which is found by join- 
 ing TP. If the point T were given, we would first draw 
 TQ tangent to the circle, and from the point of contact Q 
 draw the ordinate QM, intersecting the ellipse in P, the re- 
 quired point of contact; and by joining P and T we ^\ould 
 have the required tangent. 
 
 137. Tangent through a Given Point. 
 Let the tangent line be required to pass through a given 
 point {x'j y')] we shall then have the condition 
 
 y' = mx' + 7i, (a) 
 
 which, combined with (16), will enable us to determine m and 
 h. Equation (a) gives 
 
 h' = y" - 2mx'y' + m'x'\ 
 
152 PLANE ANALYTIC GEOMETRY. 
 
 which, substituted in (IG), gives 
 
 («^ - x")m' 4- 2x'y'm -\- h' - y'' = 0; 
 
 whence m = '- ~ 7^ . (18) 
 
 Co X 
 
 Since there are two values of m, two tangents to an ellijise 
 can be drawn through a given point. There are three cases 
 depending on the position of the point: 
 
 L If the position of the point is such that 
 
 ay + i'x" - a^V < 0, 
 
 the vakie of m will be imaginary. The point {x', y') will 
 then be within the ellipse. 
 
 II. If ahf + Vx'' - aW > 0, the two values will be real 
 and different. 
 
 III. If ay + b'x'' - a'b' = 0, the point (x/, y') will be 
 on the ellipse, the two tangents will coincide, and the equa- 
 tion can be reduced to the form (1(3). 
 
 138. Problem. To find the locus of the pomt from 
 loMcli two tangents to an ellij^sc mahe a right angle with each 
 other. 
 
 Let the equations of the tangents be 
 
 y = mx -\- VF -\- (i^nf; (a) 
 
 y = m'x 4- Vb' + a'm'\ (b) 
 Then the condition to be fulfilled is (§ 47) 
 
 mm' + 1 = 0. (c) 
 
 Eliminating ??i' from (b) and (c), the equation of the two 
 tangents will be 
 
 y — mx = Vb^ + «^??i'; 
 my -\- X = Va"^ -\- ¥nf. 
 
 Now, what we want is the locus of the point which is on 
 both tangents at once; that is, the locus of the point whose 
 co-ordinates satisfy both of these equations. To find the re- 
 quired locus, we must eliminate m from the equations, which 
 we do thus: 
 
THE ELLIPSE 153 
 
 Squaring and adding, wc have 
 
 ^uv + 1).;^ + (m= + l)f = {m' + l)(a' + F), 
 or x^ -{- y"^ = ft' -f b', 
 
 which is the equation of a circle whose centre is at the origin 
 and whose radius is Va^ -\- h"^. 
 
 We thus have the result: If we slide a right angh around 
 an ellipse so that its sides shall continually touch the ellipse, its 
 vertex will describe a circle whose radius is equal to the dis- 
 tance betiueen the ends of the major and minor axes. 
 
 139. Problem. A perpendicular being drawn from 
 either focus of an ellipse upon a moving tangent, it is required 
 to find the locus of the foot of the perpendicular. 
 
 Let 
 
 y = mx + V¥-\- d'm^ (a) 
 
 be the equation of the tangent. The equation of a line per- 
 pendicular to (a) and passing through the focus whose co- 
 ordinates are ae and is 
 
 ^ - - ^^c-^ - ^^^)- (^) 
 
 From (a) we have 
 
 y — 7nx = VF + cc^m^^j 
 and from (b), my -\- x = ae. 
 
 Squaring and adding, we get 
 
 (x' + y') (1 + m') = b'-\- ahn' + a'G" 
 = a'{l + m') 
 
 (since ^V -(- ^' = «'). 
 Therefore we have 
 
 ^' + f = a\ 
 
 the equation of the required locus, which is a circle described 
 on the major axis of the ellipse. The same result is obtained 
 if we draw the perpendicular from the other focus. 
 
 140. Perpendiculars from the Foci upon the Tangent. 
 Problem. To find an expression for the length of the per- 
 
154 
 
 PLANE ANALYTIC GEOMETRY. 
 
 pendicular from either focus upon the tangent to an ellipse at 
 the point {x', y'). 
 
 Let p and jt?' be the perpen- 
 diculars FQ, F'R respectively. 
 The equation of the tangent is 
 
 Vx'x + a'y'y - a/V = 0; 
 
 and since the co-ordinates of the 
 foci F and F^ are {ae, 0) and 
 (— ae, 0) respectively, we shall 
 have, by §4], 
 
 ab^ex' — a^b^ 
 
 ^ =-:7i^ 
 
 aF(ex' — a) 
 
 and 
 
 P' 
 
 Vb'x'' + ay Vb'x" -f ay 
 - aVex'- d'h' - ah\ex' + a) 
 
 + «y 
 
 vv 
 
 ay' 
 
 (19) 
 
 which are the required expressions for the perpendiculars. 
 
 Product of the Perpendiculars from the Foci upon the sam 
 Tangent, We find, by multiplication, 
 ^ _ a^i^i^ce - e'x'') 
 PP - i^x" + a'y" ~ 
 _ a'h\a^ - e'x'') 
 " l'x''-^a\a'b'-Jfx") 
 ^ a^b^a' - e'x'') 
 
 Vx" + a\a^ - o:'-") 
 _ a\\-e''){a^-e'x") 
 a\\-e'')x'''^a\a^-x''') 
 
 [since b^ = a'{l - e')] 
 __ a'(l - e'){a' - eV) 
 
 = i% (20) 
 
 an expression which is independent of the co-ordinates x^ and 
 
 2/'- 
 
 Hence: 
 
 Theorem XI. The rectangle contained by the perpen- 
 diculars from the foci upon a tangent to an ellipse is con- 
 stant and eqtcal to the square of the semi-minor axis. 
 
THE ELLIPSE. 155 
 
 For tlie ratio of tlie perpendiculars we have 
 p _ ci — ea;' 
 p' ~~ a -\- ex' 
 
 = '-r (§ 130) 
 
 Hence: 
 
 Theorem: XII. Tlie perpe7idiculars from the foci upon tlie 
 tangent have to each other the same ratio as the focal radii of 
 the point oftangency. 
 
 141. The Normal to an Ellipse, 
 
 Problem. To find the equation of the normal line at any 
 point of an ellipse. 
 
 Let x', y' be the co-ordinates of any point on the ellipse. 
 Then, by § 134, the equation of the tangent at that point is 
 
 VV 
 
 ^ + 1^ = 1' («) 
 
 V-x' , y 
 
 or y = ~,x H — ,, 
 
 <^ y y 
 The equation of a line through x' , y' and perpendicuhxr to 
 {a) is, by § 47, 
 
 y-y' = %{.^-^% (21) 
 
 which is the equation of the normal at x\ y'. 
 
 14z2. The Subnormal. 
 
 Def. That portion of the normal line intercepted be- 
 tween the point on the curve and the axis of X is called the 
 normal, and its projection on the axis of X is called the sub- 
 normal. 
 
 To find where the normal cuts the axis of X, we make 
 i/ = in the equation of the normal; then we get (see fig., 
 §136) 
 
 CX=x'{^-^\ = e'x', 
 
 Hence the subnormal 
 
 iVi¥= CM- CN 
 
 = X' — .t' 1 -J = -r,x' 
 
 \ a J a 
 = (1- e')x\ 
 
156 
 
 PLANE ANALYTIC OEOMETRY. 
 
 143. Theorem XIII. The normal at any point on a7i 
 ellipse bisects the angle co7i- 
 
 tained by the focal radii of ^ ^^^^P^-' 
 
 that point. 
 
 Proof Let us put 2p, ip', 
 the angles FFN iind F'FN 
 respectively. 
 
 By the theorem of sines, 
 we have 
 
 sin rp _ F_N^ 
 sill FNF~ FF' 
 
 sin t/j' _ F^N 
 sin FNF' ~ 'WF' 
 
 {a) 
 
 Now, FF and F^P are the focal radii whose lengths are 
 given by the equations (4) and (5), §120. Also, by §§136 
 and 142, we readily find 
 
 FN = ae — eV = e{a — ex') = er; 
 F'N = ae + e^^•' == e{a -\- ex') — er'\ 
 whence {a) gives 
 
 e sin FNF = sin i", e sin FNF' — sin ^'; 
 and then, since sin FNF' = sin FNF, we have 
 
 rp = r- 
 
 Therefore the normal P^ bisects the angle FFF', 
 
 Cor. The tangent at any point of an ellipse bisects the 
 exte7'ior angle formed by the focal radii of that point. 
 
 For if one of the focal radii, as F'F, be produced to any 
 point g, and the tangent P T be drawn, the angles F'FF, 
 FFQ are supplementary; and since NFT is a right angle and 
 PiY bisects the angle F'FF, FT also bisects the angle FFQ, 
 which is the exterior angle formed by the focal radii FF, F'F, 
 
 Eemark. If a ray of light proceed from F to any point 
 F on the ellipse, it will be reflected to F'. For this reason 
 the points P and F' are called /oa, or burning points. 
 
 The theorem just proved enables us to draw a tangent at 
 any point on an ellipse. Thus, let F be any point on the 
 curve; draw the focal radii FF, FF'; produce one of them, 
 as FF', and bisect the exterior angle thus formed by FT, 
 which is the tangent required. 
 
THE ELLIPSE. 157 
 
 EXERCISES. 
 
 OJUl 
 
 1. Show that there is a, certciiii segment of the major axis „v«>c..jl4 
 
 of an ellipse fronT^^mcn normals not coincident with thatj^**^' 
 axis maybe drawn to the ellipse, and two other segments from^-^ic^t-^ 
 which such normals cannot be drawn, and define these seg-^;;;;^ 
 ments. "^^ 
 
 2. Show that the normals from three or more equidistant 
 points on the major axis intersect the ellipse in points whose 
 abscissae are in arithmetical progression. 
 
 3. Show that the ordinate of the point in which a normal 
 
 intersects the minor axis is in the constant ratio ^ to 
 
 e' — 1 
 
 that of the point where it intersects the ellipse. 
 
 Reciprocal Polar Relations. 
 
 144. Chord of Contact. 
 
 Def. The line which passes through the points where 
 two tangents from an external point meet an ellipse is called 
 the chord of contact. 
 
 Problem. To find the equation of the chord of contact. 
 
 Let (7i, h) be the co-ordinates of the point from which the 
 two tangents are drawn; (a:', ?/'), the co-ordinates of the point 
 where one of the tangents through (7^, h) meets the curve, 
 and (a;", y") the co-ordinates of the point where the other 
 tangent meets the curve. 
 
 The equation of the tangent at (a:', y') is 
 
 x'x y'y _ 
 
 -^+-^-1, (a) 
 
 and since this passes through (h, k), we have 
 hx' hy' _ 
 
 Similarly, __ _^ _^- ^ 1. (c) 
 
 Hence it follows that the equation of the chord of contact is 
 «=• + J. - i, (^^) 
 
158 
 
 PLANE ANALYTIC GEOMETRY. 
 
 for tliis is tlie equation of a straight line, and is satisfied for 
 X — x', y = y' and x = x" , y = i/", as we see from (b) and 
 (c). 
 
 Cor. From what has been shown in the preceding section, 
 it is evident tliat this equation, referred to any pair of conju- 
 gate diameters as axes, is 
 
 hx ky 
 
 1. 
 
 (23) 
 
 145. Locus of Interseciion of Two Tangents. het{x%y^) 
 be the co-ordinates of any fixed 
 point Q through which the chord 
 of contact corresponding to the 
 two intersecting tangents is 
 drawn; (a;", iy"), the co-ordinates 
 of P, the intersection of the tan- 
 gents. By the preceding section, 
 the equation of the chord TT^ is 
 
 x^ y^y_ 
 a' "^ b' 
 
 1; 
 
 but since (x\ y') is a point on the chord, we have the con- 
 dition 
 
 x^'x' y"y' _ 
 
 a" '^ b-" ~ ^' 
 
 which the co-ordinates of tlie point of intersection must always 
 satisfy. Hence, regarding ic", «/" as variables and omitting 
 the accents, the equation of the locus of the point of intersec- 
 tion of the two tangents is 
 
 
 (24) 
 
 Cor, This equation, referred to a pair of conjugate diam 
 eters as axes, will be 
 
 a'' 
 
 -t- ^,2 
 
 (25) 
 
 146. Pole and Polar. 
 
 The identity of form in the equations of the iangenf, the 
 chord of contact and the locus of the intersection of tangents 
 
THE ELLIPSE. 159 
 
 drawn from the extremities of chords passing through a fixed 
 point is only the expression of a reciprocal relation whicii 
 exists between the locus and the fixed point (a?', t/'). This 
 relation is one of polar reciprocity and is expressed by the 
 following theorem: 
 
 Theorem XIV. 1. If chords in an ellipse he draiun through 
 any fixed point and tangents he drawn from the extremities of 
 each chord, the locus of the intersections of the several pairs 
 of tangents will he a straight line. 
 
 2. Conversely, If from different points in a straight line 
 pairs of tangents he drawn to an ellipse, their chords of con- 
 tact will intersect in 07ie point. 
 
 Defs. The straight line which forms the locus of the 
 intersection of two tangents drawn from the extremities of 
 any chord which passes through a fixed point is called the 
 polar of that point. 
 
 Reciprocally, the fixed point is called the pole of the 
 straight line which forms the locus. 
 
 Thus, if P be the fixed point through which the chords 
 GG', HH' are drawn, and pairs of tangents GR, G'R, HQ, 
 H' Q be drawn from their extremities, intersecting in R and 
 Q respectively, then the line QR is the polar of P, and P is 
 the pole of QR. If the polo is 
 on the curve as at H, then the 
 tangent ^i? is the polar; and if 
 the pole is without the curve, as 
 at Q, then it follows that the 
 chord of contact HH' is the, 
 polar; hence we see that the 
 tayigent and the chord of con- 
 tact are respectively the polars of the point of contact and of 
 the intersection of the tangents drawn from the extremities 
 of the chord of contact. 
 
 Hence it follows that if {x', y') be the co-ordinates of any 
 point within, on or without the curve, the equation of the 
 polar is 
 
 $ + ^^^ = 1, (26) 
 
160 PLANE ANALYTIC GEOMETRY. 
 
 or, when referred to a pair of conjugate diameters as axes, 
 
 a'' + f^n - 1. (^7) 
 
 The equation of the diameter conjugate to that which 
 passes through the point (:c', y') is 
 
 ^ + ff = 0, 
 
 which shows that the diameter and the polar (27) are parallel; 
 hence we have the following theorem: 
 
 The polar of any poi7it in respect to an ellipse is parallel to 
 the diameter conj^igate to that luhich passes through the point, 
 
 147. Polar s of Special Points. 
 
 Polar of the Centre. If in the equation of the polar (26) 
 we suppose the pole {x', y') to approach the centre, x' and y' 
 will approach zero as their limit, and one or both the co- 
 ordinates, X and y, of any point of the polar will increase 
 indefinitely. Hence tlie polar of the centre is at infinity. 
 
 This is also seen from the fact that tangents at the ex- 
 tremities of any diameter meet at infinity. 
 
 Polar of a Point on o?ie of the Axes. When y' = 0, we get 
 
 X = —,-= 'd constant, 
 
 which shows that the semi-major axis is a mean proportional 
 between the distances, a;' and x, of the pole and polar from 
 the centre. Since the same reasoning may be applied to a 
 point on the minor axis, we conclude: 
 
 Theorem XV. Either semi-axis is a mean proportional 
 ietiveen the distances cut off from it hy a pole upon it, and ly 
 the corresponding polar. 
 
 Polar of the Focus, Substituting for {x', y') the co-or- 
 dinates of either focus (± ae, 0) in (26), we have 
 
 x= ± -' 
 
TEE ELLIPSE. 
 
 161 
 
 or, the polar of either focus of an ellipse is perpendimdar to 
 the major axis and at a distance from the centre equal to — 
 measured on the same side as the focus. 
 148. Directrix of an Ellipse. 
 
 If DR is the polar of the focus F, we have 
 a 
 
 and 
 
 00 = 
 DP 
 
 OM 
 
 00- MO 
 
 a 
 
 _ a 
 ~ e 
 
 ex 
 
 but from the linear equation of the curve we have 
 
 hence 
 
 DP = 
 
 FP 
 FP 
 
 FP 
 
 and -^- = 
 
 The same reasoning applies to either focus and its polar. 
 Hence: 
 
 Theorem XVI. The focal distance of any point on an 
 ellipse is in a constant ratio to its distance fj^o^n the polar of 
 the corresponding focus, the ratio being less than unity and 
 equal to the eccentricity of the curve. 
 
 Def The polar of either focus is called a directrix. 
 
162 PLANE ANALYTie OEOMETRT. 
 
 EXERCISES. 
 
 1. Show that an ellipse has a pair of equal conjugate 
 diameters whose direction coincides with the diagonals of 
 the rectangle on the axes. 
 
 2. Show that the equal conjugate diameters of an ellipse 
 bisect the lines joining the extremities of the axes. 
 
 3. Find the co-ordinates of the point in an ellipse such 
 that the tangent there is equally inclined to the axes. 
 
 a' K' 
 
 Ans. 
 
 4. If r and r' denote the focal radii of any point on an 
 ellipse whose eccentric angle is cp, show that 
 
 r = «(1 — e cos (p) and r' = a(i -\- e cos cp). 
 
 5. Find the equation of the tangent at the extremity of 
 the latns rectum. A?is. y -\- ex = a. 
 
 6. Find the equations of the lines joining (1) the extremi- 
 ties of the axes; (2) the centre and the extremities of the latera 
 recta. 
 
 Ans.y=±-[x^a)- .V - ± -. • -• 
 
 7. Find the equation of the normal at the extremity of 
 the latus. rectum. 
 
 X 
 
 Ans, y — — 4- ae'^ = 0. 
 
 8. If the normal at the extremity of the latus rectum 
 passes through the extremity of the minor axis, show that the 
 eccentricity of the ellipse is determined by the equation 
 
 e'-{-e' -1 = 0. 
 
 9. Show that the equation of the tangent at any point is 
 
 X 11 
 
 — cos <z? 4- 4 sin q> — 1 = 0. 
 a b 
 
 where cp is the eccentric angle of tliat point. 
 
THE ELLIPSE. 163 
 
 10. Find the equation of the straight line which is tan- 
 gent to the ellipse 20?/^ + bx" = 100 at the point (2, 2). 
 
 11. Tlirough the right-hand focus of the ellipse 
 25y^ -{- IGo;" = 1600 is drawn a focal radius making an angle 
 of 30° with the axis of X. Find the equation of the tangent 
 to the ellipse at the end of this radius. 
 
 12. Express the intercepts which the normal to an ellipse 
 cuts off from the co-ordinate axes in terms of the principal 
 axes of the ellipse and of the co-ordinates of the point (x^, y^ 
 in which the normal cuts the ellipse. 
 
 13. If d is the angle which a radius from the centre of the 
 ellipse forms with the axis of X, and 6' the angle which the 
 tangent to the ellipse at the end of that radius forms with the 
 same axis, find what relation exists between 6 and 6'. ^» ^ f> ^^-' P- 
 
 14. From the centre of an ellipse to a tangent is drawn a 
 line parallel to the focal radius of the point of tangency, and 
 meeting the tangent at the point p. Find the locus of p ^^ 
 the tangent changes its position. 
 
 15. From one focus of an ellipse a perpendicular is dropped 
 upon the tangent and produced to an equal distance on the 
 other side. Show that its terminus is in the same straight 
 line with the point of tangency and the other focus, ^r C^. ^jf/A.^^ 
 
 16. The same thing being supposed, find the locus of p 
 when the tangent moves around the ellipse. 
 
 17. To the ellipse a^y"^ + 5V = a^W and its circumscribing 
 circle ?/' -j- 2;' = c^ tangents are drawn such that the points of 
 tangency shall have the same abscissa. What relation exists 
 between the subtangents, and what relation between the sub- 
 normals? 
 
 18. Find the equations of the tangents drawn from the 
 point (0, 8) to the ellipse whose equation is 20?/^ -|- hx^ = 100. 
 
 19. If that point of an ellipse to which a normal is drawn 
 approaches indefinitely near to the major axis, what limit will 
 the intercept of the normal upon the axis of X approach? 
 
 20. On the major axis of an ellipse a point is taken whose 
 abscissa is a/. Find the slope and equation of the tangents 
 from this point. 
 
164 PLANE ANALYTIC GEOMETRY. 
 
 21. At what points will the tangents which make an angle 
 of 45° wich the principal axes cut those axes? 
 
 22. Find the intercept upon the minor axis when the 
 normal approaches the end of that axis. 
 
 23. Find the equations of the two tangents to the ellipse 
 hy"^ -\- Zx" — lb which are parallel to the line 3?/ — 4.r -j- 1 =0. 
 
 24. To the ellipse 36?/' + 25^:' = 900 are to be drawn tan- 
 gents cutting the axis of X at an angle of 30°. Find the co- 
 ordinates of the points of tangency. 
 
 25. Having given the ellipse J/x" -\- a^y"^ = a'y' and the 
 circle x^ -\- y"^ =i ab, it is required to find the equation of the 
 common tangent to the two curves. Find also the angle at 
 which the curves intersect. 
 
 26. If two points as poles be taken on a tangent to an 
 ellipse, where will their polars intersect? 
 
 27. The chord of contact to two tangents of an ellipse is 
 required to pass through the focus. AVhat is the locus of the 
 point where the tangents intersect? 
 
 28. Find the pole of the line y — mx + li with respect to 
 the ellipse ay + Ux^ = a^y^, 
 
 29. If tangents to the circumscribed circle of an ellipse be 
 taken as polars, what will be the locus of the pole? 
 
 30. Find the locus of the pole w^hen the polar is required 
 to be a tangent to the circle described upon the minor axis of 
 the ellipse as a diameter. 
 
 31. If a series of poles be taken on the diameter of an 
 ellipse, show that the polars will all be parallel to each other. 
 
 32. If chords be drawn from any point of an ellipse to the 
 ends of either principal axis, show geometrically that they are 
 parallel to a pair of conjugate diameters. 
 
 33. If a line of fixed length slide with its two ends con- 
 stantly upon the respective sides of a right angle, show that 
 any point upon it describes an ellipse. 
 
 34. The area of an ellipse is to be equal to that of the con- 
 centric circle passing through its foci. Find its eccentricity. 
 
 . ) 1/5-1)4 
 
THE ELLIPSE. 165 
 
 35. The minor caxis of an ellipse is 12, and its area is equal 
 to that of a circle whose diameter is 20. What is its major 
 
 axis 
 
 36. The area of an ellipse is equal to that of a circle cir- 
 cumscribed around the square upon its minor axis. Find the 
 angle whose sine is the eccentricity. Ans. 60°. 
 
 37. Show that the equation of the normal at the point 
 whose eccentric angle is cp is 
 
 ax sec cp — hy cosec cp = a^ — If, 
 
 38. If cp and q)' be the eccentric angles of any two points 
 P , Q on an ellipse, sho\7 that the area of the parallelogram 
 formed by tangents at the extremities of the diameters 
 
 through P and Q is -^—, — -. r- When is this area a 
 
 ° ^ sin(<p' — cp) 
 
 minimum? 
 
 39. Show that the circle described on any focal chord as a| r 
 diameter touches the circle described on the major axis as aj 
 diameter. _i- 
 
 40. Normals are drawn to an ellipse and J;he circumscrib- 
 ing circle at points having the same abscissa. Show that the 
 locus of their intersection is a circle whose radius is a -\-h. 
 
 41. Show that the locus of the intersection of tangents 
 to an ellipse at the extremities of conjugate diameters is an 
 ellipse. 
 
 42. Show that the tangents at the extremities of any chord 
 of an ellipse meet on the diameter which bisects that chord. 
 
 43. If q) and q)' denote the eccentric angles of the vertices 
 of two conjugate diameters of an ellipse, show that 
 
 tan (p tan (p' -\-l = 0. 
 
 44. If 6 denote the angle which any focal chord makes 
 with the major axis, show that the length of the chord is 
 
 -TT. J r3T> and the length of the diameter parallel to the 
 
 w( L 6 cos t/ ) 
 
 , , . 2h 
 
 chord is --J- ^ ^-^. 
 
 i/(l — e'cos'^) 
 
 45. If cp and cp' be the eccentric angles of any two points 
 
166 PLANE ANALYTIC GEOMETRY. 
 
 on an ellipse, show that the equation of the chord which joins 
 the points is 
 
 Z>cosi(«^+ cp').x-^a^mi{cp^ cp') . y — abcosi((p — cp'). 
 
 46. Find the polar equation of the ellipse (1) when the 
 centre is the pole, and (2) when the left-hand vertex is the 
 pole, the major axis being the initial line in both cases. 
 
 h' %aV cos 6 
 
 A'/is. r = 5 5v.; r = 
 
 e' COS'S' a'sm'd-^d'cos'O 
 
 47. Show that the perpendicular from the centre on the 
 chord which joins the extremities of two perpendicular diam- 
 eters of an ellipse is of constant length. 
 
 48. Find the polar co-ordinates of that point on an ellipse 
 at which the angle between the radius vector and tangent is 
 a minimum. A7is. a; cos~^e. 
 
 49. If the equation x"^ -\- y^ = a^ represent an ellipse, ex- 
 press its eccentricity in terms of the angle between the axes. 
 
 50. Show that the sum of the reciprocals of two focal 
 chords at right angles to each other is constant and equal to 
 
 g' + y 
 2ab' ' 
 
 51. A tangent is inclined to the major axis of an ellipse at 
 an angle 6. Show that the rectangle contained by perpendi- 
 culars upon it from the ends of the major axis varies as cos'0. 
 
 52. If 7\, r, 1\ be the radii vectores corresponding to the 
 angles (9 - 60°, ^, ^ + 60°, show that 
 
 111 1 
 
 r, r^ r -J latus rectum* 
 
 53. Show from the equation ?/' = -J(2ax — x^) and from 
 
 § 119 that if the major axis of an ellipse becomes infinite 
 while the parameter remains finite, the ellipse will become a 
 parabola. 
 
 54. Show that the line from the focus to the point of in- 
 tersection of two tangents bisects the angle formed by the 
 focal radii of the points of tangency. 
 
CHAPTER VII. 
 THE HYPERBOL 
 
 Equation and Fundamental Properties of 
 the Hyperbola. 
 
 149. Def. An hyperbola is the locus of a point the 
 difference of whose distances from two fixed points is con- 
 stant. 
 
 The two fixed points are called the foci of the hyper- 
 bola. 
 
 The distances from any point on the curve to the foci are 
 called focal radii, oy focal distances. 
 
 The hyperbola is described mechanically as follows: Take any two 
 fixed points, as i^and F', and at 
 one of them, as F', let a ruler be 
 pivoted, while to the other point, 
 F, is fastened a thread whose 
 length is less tlian that of the 
 ruler. 
 
 Attach the other end of the 
 thread to the free end of the ruler 
 at J), and stretch the thread tightly 
 against the edge of the ruler with 
 a pencil-point, P. Then, while the ruler is moved round the pivot at F\ 
 let the pencil-point slide along the edge of the ruler so as to keep the 
 t-hread lightly stretched; the pencil-point will describe an hyperhola, be- 
 cause in every position of P we shall have 
 
 F'P - FP= {F'P + PB) - (FP -f PD). 
 
 But F'P-\- PD is the length of the ruler, and FP -\- PD is the length 
 of the thread, and the difference between the lengths of these is con- 
 stant; therefore we have 
 
 F'P - FP=a. constant, 
 which agrees with the definition. 
 
168 
 
 PLANE ANALYTIC GEOMETRY. 
 
 By interchanging the fixed extremities of the ruler and thread we 
 shall obtain a second figure equal and similar in every respect to the 
 first, but turned in the opposite direction. Thus we see that the com- 
 plete curve consists of two branches, as represented above. 
 
 150. Problem. To find the equation of the hyperlola. 
 
 Let tlie straight line drawn 
 through the foci" ^e taken 
 as the axis of X; t. "* point 
 C midivay between tj ioci 
 be taken as the origin, and 
 tlie perpendicular to FF' 
 through C as the axis of Y, 
 Let the distance between the 
 foci = 2c; the difference be- 
 tween any two focal radii = %a\ and x, y, the co-ordinates of 
 any point P. Then we have 
 
 and therefore 
 
 F'M ^ x-^-c, 
 FM =x-c', 
 
 (1) 
 
 ^^" = (^ + ^r + !/'; 
 
 PF^={x-cY^y^', 
 and, by the fundamental property of the curve, 
 
 V(^ + cf 4- y' - f (:,• - cy + y' = 2a. 
 Freeing this equation of surds, we have 
 
 (c' - a')x' - a'y' = a\c' - a'), 
 which is the required equation. 
 
 This, however, may be simplified by putting, for the sake 
 of brevity, 
 
 c'-a' = b'; (2) 
 
 hence we have 
 
 h'x' - ay = a'b\ (3) 
 
 or, dividing through by a'^»', 
 
 which is the equation of the hyperbola referred to its centre 
 and axes. 
 
THE HYPERBOLA. 
 
 169 
 
 151. Relations among Axes and Foci of the H}ipcr1)ola. 
 If, in the equation (3), we put y = 0, we have, for the 
 jioints in which the curve cuts the axis of X, 
 
 x= ±a= CA or CA'. 
 
 Therefore the curve cuts the axis of Xin two points, A 
 and A', equidistant from tlie 
 origin and between the origin 
 and the foci. 
 
 Def. The points A, A' 
 where the line joining the foci 
 cuts the curve are called the 
 vertices of the hyperbola. 
 
 The line ^^' is called the 
 transverse axis. 
 
 The point C midway between the vertices is called the 
 centre of the curve. 
 
 If X = 0, we have 
 
 y = ± b V- 1, 
 
 which shows that the curve cuts the axis of l^in two imagi- 
 nary points situated on opposite sides of the centre and at 
 the imaginary distance b V— 1 from it. 
 
 Measure off now on the axis of l^the distances CB, CB', 
 each equal to h, the real factor of this imaginary value of ?/. 
 Then: 
 
 Def. The line BB^ is called the conjugate axis of the 
 hyperbola. 
 
 Solving the equation (3), for y, gives 
 
 y=±\ vi^^^\ (5) 
 
 which is real for all values of x greater than a. Hence, when 
 x'> a,y has two real values equal in magnitude but of opposite 
 signs; therefore the curve is symmetrical in reference to the 
 axis of X. 
 
 If X < a, the values of y are imaginary; therefore no point 
 of the curve lies nearer to the centre than the vertices. 
 
170 PLANE ANALYTIC GEOMETRY. 
 
 If X increases without limit in either direction, y increases 
 without limit, and therefore the curve extends indefinitely 
 both to the right and to the left of the points A, A'. 
 
 By solving the equation of the curve for x, we can easily 
 show in a similar manner that the curve is symmetrical in 
 reference to the axis of Y. 
 
 Def. The distance CF = CF' = c of each focus from 
 the centre is called the linear eccentricity of the hyper- 
 bola. 
 
 The ratio - is called the eccentricity of the hyperbola, 
 and is represented by the symbol e. 
 
 Since c' = a'' + ^% 
 we have e = = — . (6) 
 
 Hence the eccentricity of an hyperbola is always greater 
 than unity. 
 
 From (6) we find 
 
 l^a V7~^\, (7) 
 
 which expresses the semi-conjugate axis in terms of the semi- 
 transverse axis and the eccentricity; and since e > 1, l may 
 be greater or less than a. For this reason we do not use the 
 terms major and 7ninor axis as in the case of the ellipse. 
 
 Cor. By comparing the equation of the ellipse with that 
 of the hyperbola, we see that the equation of the latter may 
 be deduced from that of the former by simply writing — Z>' 
 for 4- 'b\ Hence 
 
 Any function of b i7i the ellipse ivill be converted into the 
 corresponding function in the hyperbola by merely changing 
 h into b V — 1. 
 
 153. Equilateral Hyperbola. An hyperbola in which 
 the transverse and conjugate axes are equal is called an 
 equilateral hyperbola. 
 
 From (3) we see that the equation of the equilateral 
 hyperbola is 
 
 x' -n"^ a\ 
 
TUE IIJTEIWOLA. 
 
 171 
 
 153. Def. The parameter or lalus rectum of an hy- 
 perbola is the chord through the focus perpendicular to the 
 tiansverse axis. 
 
 Theorem I. The parameter of an hyperhola is a third pro- 
 portional to the transverse and conjugate axes. 
 
 In order to find the value of the parameter or latus rec- 
 tum, we put a; = c in the equation of the curve. The equa- 
 tion of the curve may be written 
 
 .'•=*' -4 
 
 And substituting c for x and denoting the semi-parameter by 
 p, we have 
 
 whence 
 
 or 
 
 that is, 
 
 ap — Jf', 
 a : b '.'. b \ p. 
 
 (8) 
 
 Cor. The length of the semi-parameter, in terms of a and 
 e, is 
 
 p = a{e' - 1). 
 
 154. Focal Radii. 
 
 Problem. To express the lengths of the focal radii in 
 terms of the abscissa of the point 
 from which they are draiun. 
 
 Let r and r' denote the focal 
 radii of any point P whose co- 
 oi-dinates are {x, y). Then, from 
 the figure, we have 
 
 (x - aey + f 
 
 whence 
 
 = {x - aey + -^,[x' - a^) 
 
 = {x - aey -{- le' ^l){x' 
 = e^x^ — "Haex + «'; 
 r ~ ex — a. ' 
 
 «') 
 
 (9) 
 
172 
 
 PLANE ANALYTIC GEOMETRY, 
 
 In a similar manner we find 
 
 r' = ex + a. (10) 
 
 Either of these expressions, being of one dimension in a;, 
 is called the linear equation of the hyperbola. 
 
 We observe that their difference is 2a, as it should be. 
 
 155. Conjugate Hyperhola. 
 
 We will now point out the signification of the line BB', 
 whose length is 2J and which is defined as the conjugate axis 
 of the curve. It is so called by reason of the important rela- 
 tion it bears to a companion-curve, called the conjugate 
 hyperbola, whose equation we will now develop. 
 
 Let an hyperbola be described 
 about the foci G, G' situated on 
 the axis of Y, and at the same 
 distance from the centre (7 as the 
 foci F, F' oi the hyperbola which 
 we have hitherto been consider- 
 ing. Let {Xj y) be the co-ordi- 
 nates of any point Q on this new 
 curve. Then, retaining the same 
 origin and axes of reference as before, we shall have 
 
 CG = CG' = c, x = NQ and y = CJST; 
 
 therefore G'Q' = (c + yY + q:'; 
 
 GQ' = (c-yy^x\ 
 
 Let the difference between the focal radii G'Q, GQ be 2b 
 instead of 2a. Then we shall have, by the definition of the 
 curve 
 
 ^(c + yy + ^' - i^(o-yy-^x^ = ^b, 
 
 which, when freed from radicals, becomes 
 
 Z.V - (c' - b')y' = - h\c' - b'). 
 But c' -b' =:^ a'; 
 
 therefore i'x' - ay = - a'b% 
 
 or 
 
 ^ _ .^L - _ 1 
 a' b' ~ ' 
 
 (11) 
 
THE HYPERBOLA. 178 
 
 which is the equation of the companion -curve or conjugate 
 hyperbola. 
 
 If in (11) we put x = 0, 
 
 y=±b=CB or CB', 
 
 which shows that the conjugate hyperbola has its transverse 
 axis coinciding in direction and equal in magnitude to the 
 conjugate axis of the primary curve. 
 If ^ = 0, we have 
 
 x=: ±a V— 1. 
 
 But CA = a and AA^ = 2a; therefore the transverse axis of 
 the primary curve is the conjugate axis of the new curve. 
 Thus we see that what is called the co7ijugate axis of an hyper- 
 bola is in fact the tj^ansverse axis of the conjugate hyperbola. 
 
 Def. A conjugate hyperbola is one which has the 
 conjugate axis of a given hyperbola for its transverse axis, 
 and the transverse axis of the given hyperbola for its conju- 
 gate axis. 
 
 By comparing (4) and (11) we see that the equations of an 
 hyperbola and its conjugate differ only in the sign of the con- 
 stant term. Since the conjugate hyperbola holds the same 
 relation to the axis of Fthat the original does to the axis of X, 
 we may obtain the equation of the former from that of the 
 latter by simply interchanging the quantities which relate to 
 the two axes. Thus if the equation of the original hyper- 
 bola is 
 
 b'x' - ay = a'b\ 
 
 then, by interchanging 
 
 X and y, 
 a and b, 
 
 we have, after changing signs, 
 
 JV - ay = - a'b', 
 which is the equation of the conjugate hyperbola. 
 
174 
 
 PLANE ANALYTIC QEOMETRY. 
 
 156. Polar Equation of the Hyperhola, the left-hand 
 focus being the pole, and the transverse axis the initial line. 
 
 Let the angle A'F'F = 6 and PF' = r be the polar co- 
 ordinates of any point P. Then we shall have 
 
 PF' = PF" + FF'' - 2PF\ FF' cos FF'P, 
 or PF'' = r= + 4ft'e^ - 4aer cos 6, 
 Now, by the fundamental property of the curve, we have 
 PF- PF' ^'^a, 
 
 or ^v' + ^a'e' 
 
 whence we get 
 
 r 
 
 4iaer cos 6 — r = 2a; 
 a(e' - 1) 
 
 (12) 
 
 1 -i- e cos 8' 
 which is the equation required. 
 
 The polar equation may also be very readily obtained from 
 the linear equation of the curve in the same manner as in the 
 case of the ellipse. (See § 121, Ellipse.) 
 
 157. To trace the form of the curve from its polar equa- 
 tion. 
 
 In (12) let 6 = 0. Then r = a{e-\) = F'A\ As 6 
 increases from past 90°, r increases and becomes infinite 
 when 1 + e cos 6^ = 
 
 1 
 
 or when 
 
 cos 6 
 
 Thus, while 6 increases from to the angle whose cosine is 
 
 , that portion of the curve is traced out which begins at 
 
 c 
 
THE UTPERBOLA. 175 
 
 ^'and jiasses through P to an indefinite distance from the ver- 
 tex. As 6 increases from the angle whose cosine is to 180°, 
 
 G 
 
 r is negative and decreases; hence the portion P' A in the 
 lower riglit-hand quadrant is traced out. When 6 = 180°, 
 r — — rt(e+l) = F' A. As Q increases from 180° to the angle 
 
 whose cosine is in the third quadrant, r is negative and 
 
 increases numerically, and becomes indefinitely great when 
 
 cos ^ = . Thus the portion AP*' is traced out. As d 
 
 increases from that angle in the third quadrant whose cosine 
 
 is to 360°, r again becomes positive, is at first indefinitely 
 
 e 
 
 great and then diminishes until d = 360°, when r = a(e — 1) 
 = F'A', as it should. Thus the portion P'"A' in the lower 
 left-hand quadrant is traced out. 
 
 EXERCISES. 
 
 1. Prove the following propositions: 
 
 I. The distance of each focus from the centre is ae. 
 II. The distance of each focus from the nearer vertex is 
 a{e — 1), and from the farther vertex a{e + 1). 
 
 III. The distance between the vertices of the hyperbola 
 and of its conjugate is equal to that between the centre and 
 the foci. 
 
 IV. If we put e' for the eccentricity of the conjugate 
 hyperbola, we shall have 
 
 e" + e" = e''e'\ 
 
 V. The eccentricity of an equilateral hyperbola and of its 
 conjugate are each V2. 
 
 2. Find the semi-axes and eccentricity of the hyperbola 
 
 16a;' - V = 144. Ans. a = 3; Z* = 4; e = |. 
 
 o 
 
 3. Find the eccentricity and semi-parameter of the hyper- 
 bola 36a;' - 25i/' = 900. Ans. e = -^; ^ = 7.2. 
 
176 PLANE ANALYTIC GEOMETRY. 
 
 4. What is the equation of the hyperbola when the dis- 
 tance between the foci is 6 and the difference of the focal 
 radii of any point of the curve is 4? 
 
 Ans. 5x' - 4if = 20. 
 
 5. The distance from the focus of an hyperbola to the more 
 
 remote vertex is 4 and the eccentricity is — . Find the equa- 
 
 o 
 
 tion of the curve and its latus rectum. 
 
 Ans. —x"" — —li^ = 1; latus rectum = -— . 
 9 4*^ 3 
 
 6. What is the equation of the hyperbola whose transverse 
 axis is 10 and whose vertex bisects the distance between the 
 centre and the focus? Ans. ox"" — ?/' = 75. 
 
 7. The equation of an hyberbola is x^ — 4:y^ = 12. Find 
 the equation of the conjugate hyperbola and its eccentricity. 
 
 Ans. x^ — Ay^ = — 12; e = Vd. 
 
 8. If e and e' denote the eccentricity of an hyperbola and 
 its conjugate, show that 
 
 e h 
 
 9. Find the equation of the hyperbola when the left-hand 
 focus is the origin. 
 
 Ans. 77, ^ H X = e — 1. 
 
 h a a 
 
 10. Show that by multiplying every ordinate y of an 
 ellipse referred to its centre and axes by the imaginary unit 
 
 V— 1, it will be changed into an hyperbola having the same 
 axes. 
 
 11. A line parallel to the transverse axis is drawn so as to 
 intersect both an hyperbola and its conjugate. Show that tlie 
 segments contained between the two hyperbolas diminish 
 indefinitely as the line recedes indefinitely. Also, that the 
 rectangle contained by one of those segments, and by the sum 
 of the two segments, one of which is cut out of the line by 
 each hyperbola, is equal to the square upon the transverse 
 axis. 
 
THE HYPERBOLA. 
 
 177 
 
 Diameters of the Hyperbola. 
 
 158. Def. A diameter of an hyperbola is any line 
 passing through the centre. The length of a diameter is 
 the distance between the points in which it meets the curve. 
 
 Theorem II. Every diameter of an hyperhola or of its 
 conjugate is bisected hy the centre. 
 
 Proof. Let the equation of any line through the centre of 
 ail hyperbola be 
 
 y = mx. {a) 
 
 The equation of the hyperbola is 
 
 b'x' - a'y' = a'b% {b) 
 
 and the equation of the conjugate hyperbola, 
 
 b'x' - ay = - a'b\ (c) 
 
 Solving (a) and (b) for x and y, we have 
 
 ab 
 
 and 
 
 x= ± 
 
 y 
 
 Vb'-a'm' 
 mab 
 
 (d) 
 
 VU'-a'm' ) 
 And solving (a) and (c) for x and y, we have 
 
 ab 
 
 and 
 
 ic = ± 
 
 y 
 
 Va'm'- 
 mab 
 
 (^) 
 
 Va'7n'- b^ J 
 
 From {d) and (e) we see that the points of intersection of 
 the line y = mx with the hyperbola and its conjugate are at 
 equal distances on each side of the origin. Q. E. D. 
 
 . When F > a'^m'^ or m < ± -, the values of x and y in (d) 
 
 are real, which shows that the line (a) intersects the given 
 hyperbola at finite distances from the centre; while in (e) the 
 values of x and y are imaginary, Avhich shows that the line 
 (a) does not then intersect the conjugate hyperbola. 
 
178 
 
 PLANE ANALYTIC GEOMETRY. 
 
 If J' < rf'w' or w > ± -, the values of a; and y in (d) 
 
 are imaginary, wliilc in {e) they are rea?, sliowing that the 
 line does not then meet the given hyperbola, but meets the 
 conjugate at finite distances from the centre. 
 
 159. Asymptotes. If J}" —a^m^ or m = ± -, the values 
 
 of X and y in both (4) and (5) become infinite. Hence the 
 
 diameter whose slope to the transverse axis is either A — or 
 
 a 
 
 meets the hyi^erbola or its conjugate only at infinity. 
 
 Def. That diameter of an hyperbola which meets the 
 hyperbola and its conjugate at infinity is called an asymp- 
 tote of the hyperbola. 
 
 Cor. 1. The equation of the q -^/^ 
 
 asymptote CP is 
 
 y = -X, or ^-1 = 0; (13) 
 
 and of the asymptote CQ, 
 ^=--„^, or f + 1 = 0. (14) 
 
 Cor. 2. Equations (13) and 
 (14) are the equations of the diagonals of the rectangle formed 
 by the axes of the curve. Hence : 
 
 Theorem III. The asymptotes coi^icide luith the diagonals 
 of the rectaiigle contained hy the trayisverse and conjugate axes. 
 
 160. Theorem IV. The locus of the centres of paral- 
 lel chords of an hyperhola is a diameter. 
 
 The demonstration of this theorem is similar in every re- 
 spect to that of Theorem III. of the Ellipse. Substituting 
 — V for y^ in §123 of the Ellipse, we have, omitting the 
 accents on the variables, 
 
 — Ifx -\- c^my — 0, 
 or 7/ = -^— .r, (15) 
 
THE HYPERBOLA. 179 
 
 as the equation of the locus of the centres of parallel chords. 
 This is the equation of a straight line through tlie centre, and 
 is therefore a diameter of the curve. By giving m suitable 
 values, (15) may be made to represent any line through the 
 centre. Hence 
 
 Every diameter bisects some systein of parallel chords. 
 
 If m' be the slope to the transverse axis of any diameter 
 which bisects a system of parallel chords whose slope is 7n, 
 then the equation of the diameter is 
 
 
 y 
 
 = m'x. 
 
 But, by (15), 
 
 y 
 
 b' 
 am 
 
 is also the equation 
 
 of the diameter; 
 
 therefore 
 
 w' 
 
 
 or 
 
 mm' 
 
 ~ a'' 
 
 (16) 
 
 which is the relation which must hold between the slope of 
 any system of parallel chords and the slope of the diameter 
 which bisects these chords. Hence: 
 
 Theorem V. If one diameter bisects chords parallel to a 
 second diameter, the latter ivill bisect all chords parallel to the 
 former. 
 
 161. Conjugate Diameters. 
 
 Def. Two diameters are said to be conjugate to each 
 other when each bisects all the chords parallel to the other. 
 The equation of condition for conjugate diameters is, 
 
 by (16), 
 
 mm' = -^, 
 a 
 
 where m and m' denote their respective slopes to the trans- 
 verse axis. Since the second member of this equation is posi- 
 tive, m and m' must have the same signs; that is, they must 
 be both positive or both negative. Hence the angles which 
 conjugate diameters make with the transverse axis must be 
 both acAite or both obtuse. 
 
180 
 
 PLANE ANALYTIC GEOMETRY. 
 
 If 
 
 and if 
 
 m < 
 
 a' 
 
 m'> 
 
 a' 
 
 m> 
 
 b 
 
 a' 
 
 m'< 
 
 b 
 a' 
 
 m = 
 
 -|. 
 
 m' = 
 
 a 
 
 Whence it follows that the conjugate diameters of an hyperbola 
 lie on the same side of the conj^igate axis, but on opposite 
 sides of an asymptote; so that if one of two conjugates, as PP\ 
 
 meets the hyperbola, the other, QQ'y will meet the conjugate 
 hyperbola. PP' produced bisects all chords parallel to QQ' 
 in either branch of the hyperbola, and QQ\ produced if neces- 
 sary, bisects all chords drawn between the two branches of the 
 curve and parallel to PP'. 
 
 Conversely, QQ' produced bisects all chords of either 
 branch of the conjugate hyperbola parallel to PP', and PP' 
 produced bisects all chords parallel to QQ' behveen the two 
 branches of the conjugate hyperbola. 
 
 Cor, Since the chords of a set become indefinitely short 
 near the extremity of the bisecting diameter, they will coin- 
 cide in direction with the tangent at that point. Hence: 
 
 Theorem VI. The tangent to an hyperbola at the end of a 
 diameter is parallel to the conjugate diameter. 
 
 162. Pkoblem. Give7i the co-ordinates of the extremity 
 of one diameter, to find those of either extremity of the conju- 
 gate diameter. 
 
THE HYPERBOLA. 
 
 181 
 
 Let PP' and QQ' be any two 
 conjugate diameters, and {x\ y') 
 the co-ordinates of P. 
 
 The equation of CP is 
 
 y = S^^ since w = ^„ (a) 
 and the equation of QQ' is 
 
 y = -r-^ 
 
 or 
 
 2/ 
 
 
 and the equation of the conjugate hyperbola is 
 
 h'x' - ay = - a'b\ 
 Solving (h) and (c), we have, since b^x'^ — a^y^^ = aW, 
 
 (J) 
 
 («) 
 
 and 
 
 ft 
 
 163. Theorem VII. The difference of the squares of two 
 conjugate semi-dia^neters is constant and equal to the differ- 
 ence of the sqicares of the semi-axes. 
 
 Proof Let {x', y') be the co-ordinates of P (last figure). 
 Then the co-ordinates of Q will be 
 
 If the semi-conjugates be denoted by a' and h', we have 
 
 CP^ - CQ^ = (z" + /^) -{~y'' + 5-^") 
 
 h'x' 
 
 ay b'x'' - a'y' 
 
 or 
 
 b' 
 l'-^ = 0" - b' 
 — a constant. 
 
 (17) 
 
182 PLANE ANALYTIC OEOMETRT. 
 
 164. Problem. To express the angle hetween two conju- 
 gate diameters in terms of their lengths. 
 
 Let 6 and d' denote the angles which the conjugate semi- 
 diameters CP, CQ make with the transverse axis, and (p the 
 angle PCQ between them. 
 
 Then (p=e' -e 
 
 and sin cp = sin 0' cos 6 — sin 6 cos &'. (a) 
 
 Now if (x', y') be the co-ordinates of P, those of Q will be 
 
 (jy'-/)- 
 
 therefore 
 
 sin 6 = 
 
 
 cos d 
 
 x' 
 ~ a'' 
 
 
 sin (9' = 
 
 ix' 
 ' aV 
 
 cos 6' 
 
 ay' 
 ~ bb'' 
 
 Substituting 
 
 in (a), we 
 
 get 
 
 
 
 
 sin (p = 
 
 I'x'^ - aSf' 
 aha'b' ~ 
 
 ab 
 ■ a'b" 
 
 the required 
 Cor, Fr 
 
 expression, 
 om (17) we 
 
 ! have 
 
 a constant; 
 
 (18) 
 
 therefore a' and b' increase together or decrease together. 
 Hence, when each tends to coincide with the asymptote, the 
 product a'b' tends towards infinity, and sin cp tends towards 
 0; therefore the angle between two conjugates diminishes 
 Avithout limit. When the conjugates coincide with the asymp- 
 totes, each becomes infinite. 
 
 165. Theorem VIII. T7ie area of the parallelogram 
 whose sides touch an hyperbola at the ends of any pair of con- 
 jugate diameters is consta7it and eqiial to the rectatigle formed 
 by the axes of the curve. 
 
 Proof From (18) we have 
 
 4:a'b' sin (p = 4^5 
 
 = a constant, (19) 
 
 which proves the proposition. 
 
THE HYPERBOLA. 
 
 183 
 
 166. Problem. To find the equation of the hyperhola 
 referred to a2)air ofcovjvgate 
 diameters as axes. 
 
 Let DD', HH' be any pair 
 of conjugate diameters. Take 
 DD' for the new axis of X, 
 and HH' for the new axis of 
 Yy and let the angle XCD= a 
 and XCH = /?. We may 
 now transform 
 
 b'x' - ay = a'b' 
 
 from rectangular to oblique axes by the process of § 129, or 
 we may simply change b' into — b' in equation (12) of that 
 section. Thus we get 
 
 {a' shi'a - b' cos'a)x''-{- {a' shi'/3- b' cos' /3)y" 
 
 a'b\ (20) 
 
 which is the equation required. 
 
 By putting x' and y' each equal to zero, we get the inter- 
 cepts on the axes or the lengths of the semi-conjugates. 
 Thus, when y" = 0, 
 
 
 x'' 
 
 = 
 
 
 
 — 
 
 a'b' 
 
 
 = 
 
 CD' = a"; 
 
 
 a' 
 
 sii] 
 
 i'a 
 
 — b' cos 
 
 'a 
 
 and 
 
 when 
 
 x" 
 
 = 
 
 0, 
 
 we 
 
 have 
 
 
 
 
 
 /' 
 
 — 
 
 
 
 
 
 aW 
 
 
 i ~ 
 
 - CH' = 
 
 
 ^,2c 
 
 ir.2 
 
 R h^nn 
 
 ^2/; 
 
 h". 
 
 (a) 
 
 (i) 
 
 Because the new axis of Xmeets the given hyperbola, the now 
 
 axis of 1^ will not meet the curve, but will meet the conjugate 
 
 — a'b' 
 hyperbola. Therefore ^g^.^.,^ ^^^^^^^ is a negative quan- 
 
 a'sm'fi-b'cos'/3 
 
 titv. 
 
 From (a) and (b) we get 
 
 a' sin' a — b' cos' a = — 
 
 a'b' 
 
 and 
 
 a'sm'P-b'cos'P 
 
 a^ 
 ~b"' 
 
184 
 
 PLANE ANALYTIC GEOMETRY. 
 
 Substituting in (20) and dividing by — d'b"^, we have, 
 omitting the accents from the variables, 
 
 Also, the equation of the conjugate hyperbola referred to 
 tlie same axes is 
 
 X' _ f 
 
 Tangent and Normal to an Hyperbola. 
 
 167. Problem. To find the equation of the tangent to 
 an hyperbola. 
 
 In order to obtain the equation of the tangent, we have 
 only to repeat the process of §§ 134, 135, changing b"^ into 
 
 b'\ Thus we get 
 
 or 
 
 xx' 
 
 b'x'x = - a'b% 
 
 Ml -1 
 b' ~ ' 
 
 (22) 
 
 and also y = mx ± y a'm^ — ^>% (23) 
 
 where m is the slope of the tangent to the transverse axis. 
 
 Intercept of the Ta7igent 
 on the Axis of X. 
 
 In (22) make y = 0. 
 Then 
 
 ^=~ = CT, 
 
 (24) 
 
 from which we see that x 
 
 and x' must always have the 
 
 same sign; and since x is always positive in the right branch 
 
 of the curve, the tangent to that branch always intersects the 
 
 axis of X to the right of the centre. 
 
 168. Subtangent. Foj- the length of the subtangent we 
 liave, from the figure, 
 
 Subtangent = MT 
 
 
 (25) 
 
THE HYPERBOLA. 185 
 
 169. Theorem IX. The ta?igent to an hyperbola at any 
 point bisects the angle formed by the focal radii of that point, 
 
 a^ 
 Proof Since F'C=FC=ae and CT = ~, we hiive 
 
 * • "V • ■ 
 
 X' 
 
 
 rT=ae-^^, = ^-Xex'^a) 
 
 and 
 
 FT=cte-^^, = ^-,{ex'-ay, 
 
 whence 
 
 F'T ex'-\-a 
 FT ~ ex-a 
 
 
 _F'P 
 
 pp. (§154) 
 
 Therefore, since the base of the triangle F'FP is divided pro- 
 portionally to its sides (Geom.), the tangent P2^ bisects the 
 angle FPF', 
 
 170. Tangent through a Given Point. 
 
 Let (/i, ^•) be the co-ordinates of the given point, and {x' ,y') 
 the co-ordinates of the point of contact. The equation of the 
 tangent is 
 
 b'^x'x — a^y'y = ^'Z*'; 
 
 but since the tangent must pass through {li, k) and {x\ y'), 
 we have 
 
 bVix' - a'ky' = a'b% (a) 
 
 and also b'x'' - a'y" = a''b\ (b) 
 
 Eliminating y' from these equations, we have 
 
 {a'k' - bVL')x" + 2a'bVix' - a*{b' + F) = 0; 
 whence 
 
 , _ aV/h Ta'k V¥k' - b'¥ + a'b' ,^^. 
 
 "" ¥¥^ln^ • ^^^^ 
 
 Since x' has two values, two tangents to an hyperbola can 
 be drawn through a given fixed point. The tangents will be 
 real, coincident or imaginary according as 
 
 a'F -Z»7i' + a'<^'>, = or < 0; 
 that is, according as the given point is luithin, on or outside 
 the curve. 
 
186 PLANE ANALYTIC GEOMETRY. 
 
 171. Problem. To find the criterion that the ttoo tan- 
 gents from a given point shall touch the same branch of the 
 hyperbola. 
 
 If the tangents belong to the 5<ime branch of the curve, the 
 abscissae of the points of contact x' will have like signs; but 
 if they belong to opposite branches, unlike signs. Now, in 
 order that the values of x' in (26) may have like signs, we 
 must have numerically 
 
 a'bVi > a'k Va'k' - b'h' + a'b'-, 
 whence, by reduction. 
 
 k < -h, 
 a 
 
 But y = —X 
 
 *^ a 
 
 is the equation of the asymptote; and if we take on the 
 asymptote a point whose abscissa x is equal to the abscissa It 
 of the point from which two tangents may be drawn, we shall 
 have k y y\ that is, the ordinate of the point from which 
 two tangents can be drawn to the same branch of an hyperbola 
 must be less than the corresponding ordinate of the asymptote. 
 Hence the point from which two tangents can be drawn to the 
 same branch of an hyperbola must lie in the space between the 
 asymptotes and the adjacent branch of the curve, which is the 
 required criterion. 
 
 Hence, also, if the point lie without this space, the two 
 tangents will touch different branches of the curve. 
 
 173. Problem. To find the locus of the point from 
 which two tangents to an hyperbola make a right angle with 
 each other. 
 
 The solution is similar to the corresponding problem in 
 the Ellipse. We will therefore simply change b"^ to — b"^ in 
 the process of § 138, and we get 
 
 x" -\-y' = a' - b' 
 for the required locus, which is a circle having the same 
 centre as that of the hyperbola and whose radius is Va'^ — b'\ 
 
THE HYPERBOLA. 187 
 
 Cor. Two tangents at riglit angles to each other cannot 
 be drawn to an hyperbohi when b > a. 
 
 173. Problem. To find the locus of the intersection of 
 the tangent with the jjcrpendicular on it from the focus. 
 
 The sohition is the same as that of the corresponding 
 problem in the case of the ellipse. 
 
 The equation of tlie required locus is found to be 
 
 ^'' + ^' = «% 
 which is a circle described on the transverse axis as a diameter. 
 
 174. Problem. To find the length of the peiyendiciilar 
 from either focus upon the tangent to an hyperbola. 
 
 If {x', y') be the co-ordinates of the point of t^ingency, 
 and p, p' the perpendiculars from the foci F and F' respec- 
 tively, we find, in the same manner as in the Ellipse, 
 
 — ^(^'i^^' ~ ^) 
 
 , _ ab'jex'-i-a) , j ^ '^ 
 
 ^ ~ Vb'x'' 4- ay' J 
 whence we get, by reduction, 
 
 pp' = b' (28) 
 
 ,^d ^,- = ?^ =: ^, (29) 
 
 2? ex -\- a r 
 
 where r and r' denote the focal radii of the point of contact. 
 From the last two equations we readily find 
 
 y = %b\ p'' = -b' 
 
 and f = ^-. (30) 
 
 175. Normal to an Hyperbola. 
 
 Problem. To find the equation of the normal to an hyper- 
 bola. 
 
 The equation of the normal is found by changing b" into 
 — h' in the process of § 141. Thus, if (x', y') be the co- 
 
188 
 
 PLANE ANALYTIC GEOMETRY. 
 
 ordiiiates of any j^oint P on the hyperbola, the equation of 
 the normal PN is 
 
 y-y' = -i'i[''-'')' 
 
 (31) 
 
 The Subnormal. 
 
 X = 
 
 Putting y 
 
 in (31), we find 
 CN- CM= {e'-l)x\ 
 
 x' = e'x' = 
 
 Hence, subnormal = 3IN 
 
 176. Theorem X. The normal at a7iy point of an hyper- 
 bola bisects the external angle contained by the focal radii of 
 that point. 
 
 Since the angles FPF' and FP H ^yq supplementary and 
 TP bisects FPF', therefore PN, which is perpendicular to 
 TP, must bisect the external angle FPH. 
 
 Cor. Comparing this result with that of § 143, we see that 
 if an ellipse and an hyperbola have the same foci, the curves 
 loill intersect at right angles. 
 
 For at the point of intersection the tangent of one will be 
 the normal of the other, and vice versa. 
 
 Remark. The student should note the relations between the differ- 
 ent theorems and formulae relating to the ellipse and the corresponding 
 ones relating to the hyperbola. Where the formula of the one class con- 
 tains the symbol Ij^, it may be applied immediately to the other by chang- 
 ing the sign of 5^, which will be the result of substituting b V —\ for b. 
 Where only the first power of b enters, the theorems of one class involving 
 real quantities will be imaginary when transferred to the other class. Thus 
 w^e have imaginary asymptotes to the ellipse. Tlie apparent exceptions 
 arise from our substituting a real for an imaginary conjugate axis in the 
 hyperbola and thus referring several expressions which would have been 
 imaginary to the conjugate hyperbola, which, it must be remembered, 
 is not a part of the curve at all. 
 
THE HYPERBOLA. 189 
 
 Poles and Polars. 
 
 177. Pkoblem. To find the equation of the chord oj 
 contact of two tangents from the savie given point. 
 
 Let (7i, k) be the co-ordinates of the fixed point from 
 which the two tangents that determine the chord are drawn. 
 Then, by simply changing the sign of Z>' in § 144, the equa- 
 tion of the hyperbolic chord of contact will be 
 
 h'x k'y _ 
 when referred to the axes of the curve. 
 
 (32) 
 
 '$-^^1 = ^ (33) 
 
 when referred to any pair of conjugate diameters. 
 
 178. Locus of Lnter section of Two Tangeyits whose chord 
 of contact passes throicgh a fixed point. 
 
 Let {x' , y') be the co-ordinates of any fixed point through 
 which the chord of contact belonging to any two intersecting 
 tangents is drawn. Then, by simply changing the sign of y 
 in the process of § 145, we shall have, for the equation of the 
 required locus, 
 
 •'^-^-l- (34) 
 
 or, when referred to a pair of conjugate diameters, 
 
 x'x y'y _ .^.. 
 
 which is the equation of a straight line, the polar of the 
 point {x', y'). 
 
 Cor. The student may easily show, as in the case of the 
 ellipse, that the polar of any point in respect to an hyperbola 
 is parallel to the diameter conjugate to that which passes 
 through the point. 
 
190 
 
 PLANE ANALYTIC GEOMETRY. 
 
 1*79. Polar s of Special Points. 
 
 Polar of the Centre, Proceeding in the same manner as 
 in the ellipse, we find that the polar of the centre is at in- 
 finity. 
 
 Tlie Polar of any Point on a Diameter A is a straight line 
 parallel to the conjugate diameter, and cutting the diameter 
 A at a distance from the centre equal to the square of the 
 semi-diameter on which the point is taken divided by the 
 distance of the point from the centre. 
 
 Polar of the Focus. Substituting {±ae, 0) for {x/ y') in 
 the equation of the polar, we have 
 
 X = ±-. 
 e 
 
 Hence the polar of the focus of ayi hyperlola is the perpe^i- 
 
 dicular which cuts the transverse axis at a distance — from 
 
 e -' 
 
 the centre on the same side as the focus. 
 
 180. Distance of any Point on the Curve from either Fo- 
 cal Polar. 
 
 Let DRhe the polar of the focus F. Then we have 
 
 00='^; 
 
 e 
 
 DP = OM 
 
 = CM- CO 
 
 ex 
 
 FP 
 
 FP 
 
 therefore y^p = e. 
 
 Whence: 
 
 Theorem XI. The focal distance of any point on an hyper- 
 bola is in a coristant ratio to its distance from the polar of the 
 focus. 
 
THE nYPERBOLA. 
 
 191 
 
 This ratio is greater than unity and equal to the eccentri- 
 city of the curve. 
 
 Def. The polar of the focus is called the directrix of 
 the hyperbola. 
 
 The above property enables us to describe the curve by continuous 
 motion, as follows: Take any fixed 
 straight line NR and any fixed point F, 
 and against the former fasten a ruler, 
 and place another ruler, right-angled at 
 N, so that its edge, NH, may move freely 
 along NR. At F attach one end of a 
 thread equal in length to the hypothe- 
 nuse HQ of the ruler, and the other end 
 to the extremity Q of the ruler. Then 
 with a pencil-point P stretch the thread 
 tightly against the edge HQ, while the 
 ruler is moved along the other ruler, iViJ. 
 The point P will describe an hyperbola, 
 for in every position we shall have 
 
 and therefore 
 
 = a constant. 
 
 PF= PR; 
 
 PF _ PR 
 PI)'~ PD 
 
 _m 
 
 NQ 
 
 181. Cor. From §§ 97, 148 and 180 it follows that we 
 may define a conic section as the locus of a point which moves 
 in such a way that its distance from a fixed point (the focus) 
 is in a constant ratio to its distance from a fixed straight 
 line (the directrix). 
 
 PF 
 
 In the ellipse, the ratio -p-^ 
 
 PF 
 In the parabola, the ratio -p^ 
 
 PF 
 
 In the hyperbola, the ratio -pj: 
 
 < 1. 
 
 = 1. 
 
 > 1. 
 
 In all cases this ratio is the eccentricity of the curve. 
 
 In the case of the ellipse and hyperbola there is a direc- 
 trix corresponding to each focus. In the case of the parabola 
 the second focus and directrix are at infinity. 
 
192 PLANE ANALYTIC GEOMETRY. 
 
 The Asymptotes. 
 
 183. We have already shown (§159) thit the equations 
 of the asymptotes when re- 
 ferred to rectangular co-ordi- ^ 
 nates are 
 
 a b 
 
 Now since the equation of 
 the hyperbola referred to a 
 pair of conjugate diameters 
 as axes is of the same form as -^^ ^ \^H 
 
 when referred to rectangular axes, we at once infer that equa- 
 tions {a) transformed to the same conjugate diameters become 
 
 a' "^ b' ~ ^' 
 
 that is, the equations of the asymptotes MG, LH when re- 
 ferred to any pair of conjugate diameters are respectively 
 
 and i^+t' = °- (") 
 
 Equation {h) is the equation of a line which passes through 
 the centre or origin and the point {-{- a', -\- h')', that is, 
 through C and 2); and (c) is the equation of a line which 
 passes through the origin and the point (-f a', — h') or C 
 and E. Hence we conclude: 
 
 Theorem XI. The asymptotes comcide in direction with 
 the diagonals of the parallelograin formed ly any pair of 
 conjugate diatneters. 
 
 183. Angle between the Asymptotes. 
 
 Let GCX=a. Then tan « = -; 
 
 a 
 
 b 
 whence sin a = — , and cos a 
 
 X 
 
 — 
 
 y 
 b' 
 
 X 
 
 + 
 
 y 
 
 Va' + b' Va' + b' 
 
THE HYPERBOLA. 193 
 
 Now since XCH = GCX, sin GGH=. sin %GCX', 
 hence sin GGH = 2 sin G^CXcos GGX 
 
 a' + b'' 
 
 Cor. In the equilateral hyperbola, a = h; 
 
 hence sin GCH — 1; 
 
 that is, the asymptotes of the equilateral hyperbola intersect 
 at right angles. For this reason the equilateral hyperbola is 
 sometimes called the rectangular hyperbola. 
 
 184. Theorem XII. Tlie asymptotes of the hyperMa 
 are its tangents at infinity. 
 
 We prove this by showing that, as the point of tangency 
 on an hyperbola recedes indefinitely, the tangent approaches 
 the asymptote as its limit. 
 
 1. If, in equation (24) of § 167, 
 
 _a' 
 
 we suppose x' to increase without limit; x, the abscissa of the 
 point in which the tangent intersects the transyerse axis, ap- 
 proaches zero as its limit. Hence the tangent at infinity 
 passes through the centre of the hjrperbola. 
 
 2. From the equation of the tangent, 
 
 h'^x'x — a'y'y — a^V, 
 
 Vx' 
 it follows that its slope to the axis of X is -^-,. We must 
 
 asy 
 
 now find the value of this slope when the point of tangency 
 {x* , y') recedes to infinity. Because this point remains on 
 the hyperbola, we have 
 
 x^ _ ir _ 
 a' b^ ~ ^' 
 
 whence 
 
 r: =/«+#)■ 
 
194 PLANE ANALYTIC GEOMETRY. 
 
 As y' recedes to infinity,— 5 approaches zero as its limit; whence, 
 
 3^' (I 
 
 at infinity,--^ = ± -p and we have 
 
 Slope of tangency at infinity = a: -. 
 
 Hence the tangents at infinity are a pair of lines whose equa- 
 tions are 
 
 y=±-x, 
 
 which lines are the asymptotes, by definition. 
 
 185. Problem. To find the equation of the hyperlola 
 referred to its asymptotes as axes. 
 
 Let the asymptote CH be the new axis of X, and the 
 other, CG, the new axis of Y\ {x, y) be the co-ordinates of any 
 point on the curve referred to the old axes, and (x', y') the 
 co-ordinates of the same point referred to the new axes. 
 
 The equation of the curve referred to the old axes is 
 
 h'x' - aY = cc'b\ (a) 
 
 which must be transformed to the new or oblique axes, the 
 origin remaining the same. 
 
 The formulae of transformation are 
 
 X = x' cos a -{- y' cos /?; ) /^x 
 
 y = x' sin a -\- y' sin /?; f 
 
 where a and /? are the angles which the new axes make with 
 the old axis of X\ that is, a = XCH and ^ = GCX, 
 or /3 = — a. 
 
 Therefore (b) becomes 
 
 X = (x' -\- ?/')cos a; 
 
 y = {x' — ?/')sin «; 
 
 which being substituted in {a) give, after obvious reductions, 
 
 (^'cos'o:- a" sin'a)(a;"+?/")+ ^b'' cos'a'+ a^ sin' a)x'y'=a''b\ 
 
 -D , ^ b sin a b 
 
 But tan a = —, or = -: 
 
 a cos a a 
 
 whence b^ cosV = «' sin'<a'; 
 
THE HYPERBOLA. 195 
 
 which being substituted in the preceding equation gives, after 
 dividing by a^ 
 
 4 sin'or x'y' = Z>'. 
 But sin a = - . ^ ; (§ 183) 
 
 therefore 4a;'?/' = a"* + Z>% 
 
 or, omitting the accents on the variables, since the equation 
 is perfectly general, 
 
 ^y = '^. (36) 
 
 which is the required equation. 
 
 Cor. The equation of the conjugate hyperbola referred 
 to the same axes is readily found to be 
 
 -^ = -^'. m 
 
 186. Problem. To find the equation of the tangent to 
 an hyjwrhola referred to the asymptotes as axes. 
 
 Let {x', y') and (re", «/") be the co-ordinates of any two 
 points on the curve. The equation of the secant through 
 these two points is 
 
 y -y' = l^ 1 1^ - ^'). («) 
 
 Since the points (x^, y') and {x", y") are on the curve, 
 
 a' + l>" 
 
 ^y = 
 
 and X y 
 
 .,.,. _«' + *' 
 
 //.i' 
 
 whence x'y' = x"y", or y" = —j-,- 
 which substituted in (a) gives, after reduction, 
 
 y -y' = - ^(^ - ^')- 
 
196 
 
 PLANE ANALYTIC GEOMETRY. 
 
 Now at the limit, x" = x' and the secant becomes a tan- 
 gent; hence the equation of the tangent at the point {x\ y') is 
 
 y 
 
 whence 
 
 or 
 
 -y'--=- 
 
 
 x'y -\- xy' 
 
 = 3^'y', 
 
 x' ^ y' 
 
 -2, 
 
 (38) 
 
 which is the simplest form of the required equation. 
 
 Co7\ 1. Making x and y successively equal to 0, we get the 
 intercepts on the axes; 
 
 thus, x=:2x' = CT 
 
 and y = 2y'= CT, 
 
 Hence the point of contact 
 is the middle point of TT'\ 
 or, that portion of a tan- 
 gent intercepted between 
 the asymptotes is bisected 
 at the point of contact. 
 
 Cor. 2. CT X Cr = 4:x'y'= a' + h'; 
 
 or, the rectangle formed by the intercepts cut off by any tan- 
 gent from the asymptotes is constant and equal to the sum 
 of the squares of the semi-axes. 
 
 Cor. 3. The area of the triangle CTT' is 
 
 = iCT.Cr . sin Tcr 
 
 ^ a" -{- b" 
 a' + i' 2ab 
 
 2 • a^ + b' 
 = ab, a constant ; 
 
 or, the area of the triangle formed by any tangent and the 
 asymptotes is constant and equal to the rectangle of the 
 semi-axes. 
 
THE HYPERBOLA. 197 
 
 EXERCISES. 
 
 1. Find the equation of that hyperbola whose transverse 
 axis is 8 and which passes through the point (10, 25). 
 
 Ans. zrx — ■' — ^ 
 
 16 2500 
 
 2. What condition must the eccentricity of an hyperbola 
 fulfil in order that the abscissa of some point upon it shall be 
 equal to the ordinate? A?is. e < V2. 
 
 3. Express the distance from the centre of an hyperbola to 
 the end of its parameter in terms of the semi-transverse axis 
 and eccentricity. 
 
 4. Show that each ordinate of an equilateral hyperbola 
 is a mean proportional between the sum and difference of the 
 abscissa and semi-transverse axis. 
 
 5. Write the equation of a focal chord cutting an hyperbola 
 at the point (ic', y'). 
 
 6. Find that point upon the conjugate axis from which the 
 two tangents to an hyperbola form a right angle with each 
 other. 
 
 7. Where do the tangents drawn from a vertex of the con- 
 jugate hyperbola touch the hyperbola, and what are the equa- 
 tions of these tangents? Show that they are bisected by the 
 transverse axis. 
 
 8. What must be the eccentricity in order that the tan- 
 gent at the end of the parameter may pass through the vertex 
 of the conjugate hyperbola? 
 
 9. Find those tangents to an hyperbola which make an 
 angle of 60° with the transverse axis. 
 
 10. What must be the eccentricity of an hyperbola that the 
 subnormal may always be equal to the abscissa of the point 
 from which the normal is drawn? 
 
 11. Find the equation of the hyjierbola when the origin is 
 transferred to one of the vertices, while the axes of co- 
 ordinates remain parallel to the principal axes. 
 
 12. Express the product of the segments into which a 
 
198 PLANE ANALYTIC GEOMETRY. 
 
 focal chord is divided by the focus in terms of the angle 
 which the chord forms with the major axis. 
 
 Ans. 
 
 1 - e'^cos''^ • 
 
 13. Show that the sum of the reciprocals of the two seg- 
 ments of a focal chord is equal to four times the reciprocal of 
 the parameter. 
 
 14. The line x = dy \s n diameter of the hyperbola 
 25x' — 16y' = 400. Find the equation of the conjugate 
 diameter. 
 
 15. For what point of an hyperbola are the subtangent and 
 subnormal equal to each other? 
 
 16. Express the length of the tangent at the point {x', y'). 
 
 17. Find the condition that the line -^ -f ^ = 1 shall touch 
 the hyperbola -^ — -^ = 1. Ans. e" — e^ — 1. 
 
 18. A perpendicular is drawn from the focus of an hyper- 
 bola to an asymptote. Show that its foot is at distances a 
 and h from the centre and focus respectively. 
 
 19. Show that the linear equation of the right-hand branch 
 of the hyperbola when a focus is the origin is 
 
 r =: ea; =F «(1 — e')- 
 
 20. Each ordinate of an hyperbola is produced until it is 
 equal to the focal radius of the point to which it belongs. 
 Find the locus of its extremity. 
 
 21. Find the equation of the tangent at the extremity of 
 the latus rectum. 
 
 22. Show that the intercepts cut off from the normal by 
 the axes are in the ratio of ct^ : 5^ 
 
 23. In an hyperbola, 3fl = 2c. Find the eccentricity and 
 
 the angle between the asymptotes. 
 
 A 3 . _,4 .- 
 
 Ans. e = — ; sm ^ — r 5. 
 z y 
 
 24. Show that the angle between the asymptotes of an 
 
 hyperbola is 
 
 2 sec~^ e. 
 
THE HYPERBOLA. 199 
 
 25. From any point on an hyperbola perpendiculars are 
 drawn to the asymptotes. Show that their product is constant 
 
 and equal to -^— - — t^. 
 ^ a -{- b 
 
 26. From any point in one of the branches of the conju- 
 gate hyperbola tangents are drawn to an hyperbola. Show 
 that the chord of contact touches the other branch of the 
 conjugate hyperbola. 
 
 27. Show that the polar equations of the right-hand branch 
 of an hyperbola referred to the foci are 
 
 r — —^ ^ and r = —^ ^. 
 
 1 — e cos a 1 — e cos 6 
 
 28. Show that the polar equation of the hyperbola when 
 the centre is the pole is 
 
 ,^ ^ 
 
 ^ e' cos" 6-1' 
 
 29. Show that the length of any focal chord of an hyper- 
 
 2 y^ 
 
 bola is - . -5 ^-7^ -, where 6 is the inclination of the 
 
 a e cos 6—1 
 
 chord to the transverse axis. 
 
 30. In the figure of § 182, show that the diagonal P(> is 
 parallel to the asymptote. 
 
 31. In an equilateral hyperbola, if cp is the inclination of 
 a diameter passing through any point P, and ^' the inclina- 
 tion of the polar of P, show that 
 
 tan cp tan q)' = 1. 
 
 32. Through the point (5, 3) is to be drawn a chord to the 
 hyperbola 25a:'' — IQy^ =z 400 which shall be bisected by the 
 point. Find the equation of the chord. 
 
 33. In an hyperbola is to be inscribed (or escribed) an equi- 
 lateral triangle, one of whose vertices shall be at the right- 
 hand vertex of the curve. Find the sides of the triangle, 
 and find the eccentricity when they are infinite. 
 
 34. Express the tangent of the angle between the two 
 focal radii drawn to the point (^', «/') of an hyperbola, and 
 
200 PLANE ANALYTIC GEOMETRY. 
 
 thence find those points of the curve from which these radii 
 subtend a right angle. 
 
 Ans., m part, tan q> = -j^—, — y^ r-r. 
 
 ^ ^ a:" -[- 2/ — « e 
 
 35. For what point on an equilateral hyperbola is the pro- 
 duct of the focal radii equal to db"^? 
 
 36. From the foot of any ordinate of an hyperbola a tangent 
 is drawn to that circle described upon the major axis as a 
 diameter. Show that the ratio of the ordinate to the tangent 
 is a constant and equal to e"^ — 1. 
 
 37. Find the lengths which the directrix of an hyperbola 
 cuts off from the asymptotes, and the length of that segment 
 of the directrix contained between the asymptotes. 
 
 Ans. a and i ~- e. 
 
 38. Find the polar of the vertex of the conjugate hyper- 
 bola relatively to the principal hyperbola. 
 
 39. From any point of an hyperbola is drawn a parallel to 
 the asymptote, terminating at the directrix. Find the ratio 
 of the length of this parallel to the focal radius of the point, 
 and show that it is a constant. 
 
 40. Show (1) that the sides of the quadrilateral whose ver- 
 tices are at the termini of any pair of conjugate diameters are 
 equally inclined to the principal axes; (2) that all such quad- 
 rilaterals in the same hyperbola have their corresponding sides 
 parallel and are equal in area. 
 
 41. Find that point of an hyperbola for which the tangent 
 is double the normal. 
 
 42. At what angle does the hyperbola x" — y"" = a" inter- 
 sect the circle x^ -\- y^ = Oa'^? 
 
 43. A line drawn perpendicular to the transverse axis of 
 
 an hyperbola meets the curve and its conjugate in P and Q 
 
 respectively. Find the loci of the intersection of the normals, 
 
 and of the tangents, at P and Q. 
 
 11^ x^ Vx^ 
 Ans. The transverse axis; ~ — ~ = 4--^^. 
 
 b a^ a^y 
 
 44. The two sides of a constant angle slide along a para- 
 bola. Find the locus of the vertex of the angle, and compare 
 the cases of two loci whose angles are supplementary. 
 

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CHAPTER VIII. 
 
 THE GENERAL EQUATION OF THE SECOND DEGREE. 
 
 187. Tiie most general equation of the second degree be- 
 tween two variables x and y may be written in the form 
 
 7W.7;' + ny" + %lxy + %])x + ^y + fZ = 0; (1) 
 
 the six coefficients m, n, I, p, q and d being any constants 
 w^iatever. * 
 
 We may divide the equation throughout by any one of the 
 coefficients without changing the relation between x and y. 
 One of the six coefficients will then be reduced to unity. 
 Hence the six coefficients are really equivalent to but five in- 
 dependent quantities. 
 
 The problem now^ before us is: What possible curves may 
 be the locus of the general equation, and what common pro- 
 perties have these curves? 
 
 One property may be recognized at once by determining 
 the points of intersection of the curve with a straight line. 
 Let the equation of the straight line be 
 
 y z=zhx-\- h. 
 
 By substituting this value of y in (1) we shall have an 
 equation of the second degree in x whose roots will give the 
 abscissas of the points of intersection. Now, since an equa- 
 tion of the second degree always has two roots which may be 
 real, equal or imaginary, we conclude: 
 
 * Three of these terms are "written with the coefficient 2 because 
 many expressions which enter into the theory^ especially when deter- 
 minants are introduced, are thus simplified. We then consider I, p and 
 q as representing one half the coeflacients of xi/, x and y respectively. 
 
202 PLANE ANALYTIC OEOMETRY. 
 
 Theorem I. Every straight Ime intersects a curve of the 
 second degree in two real, coincident, or imaginary poi^its, 
 
 188. Change of Origin, To continue the investigation, 
 we change the origin of co-ordinates without changing the 
 form of the curve. If we put x' and y' for the co-ordinates 
 referred to the new origin, we must, in equation (1), put 
 
 X = x' -\- a', 
 
 y = y' + V; 
 
 a' and h' being the co-ordinates of the new origin, which are 
 to be determined in such a way as to simplify the equation. 
 Making this substitution, the equation becomes 
 
 mx'''-\- 2lx'y' + ny''' + 2(a'm + h'l + p)x'^ 2(a'l + b'n + q)y' 
 + a''7n + b''n + 2a'b'l + 2a'p -f 'Z^q + c? = 0. (2) 
 
 We now so determine the co-ordinates a' and b' that the 
 coefficients of x^ and y' shall vanish. To effect this we have 
 
 the equations 
 
 nm' + lb' =-p',) (^) 
 
 la' + nb' = -q;) ^ ^ 
 
 in which «' and b^ are the unknown quantities. Solving the 
 
 equations, we find 
 
 . _ np-lq ^^ 
 
 ^ 72 > I 
 
 I -mn [ . (3) 
 
 r — mn J 
 
 Omitting for the present the special case in which ? — mn 
 = 0, these values of a' and b' will always be finite. 
 
 By means of these values of a' and b' we may simplify tlie 
 equation (2) as follows: Multiplying the first of equations (a) 
 by a', the second by b\ and adding, we have 
 
 ma" + nb" + 2Ia'b' = - a'p - b'q. (b) 
 
 By means of the equations (a) and (J) the general equation 
 (2), omitting accents, is reduced to 
 
 mx^ + 2lxy + nf + a'p -^b'q^d^^, (4) 
 
 This equation (4) will now represent the same curve as 
 (1), only referred to new axes of co-ordinates. 
 
GENERAL EQUATION OF THE SECOND DEGREE. 203 
 
 189. A second fundamental property of the locus of the 
 second degree is immediately deducible from (4). If x and 
 2/ be any values of the co-ordinates which satisfy this equation, 
 it is evident that — x and — y will also satisfy it. That is, 
 if the point {x, y) lie on the curve, the point (— x, — y) will 
 also lie upon it. But the line joining these two points passes 
 through the origin and is bisected by the origin. When re- 
 ferred to the original system (1), this origin is the point 
 whose co-ordinates are a' and b' in (3). Hence: 
 
 Theorem II. For every curve of the second degree there is 
 a certain point ivhich bisects every chord of the curve passing 
 through it. 
 
 Def. The point which bisects every chord passing through 
 it is called the centre of the curve. 
 
 Eemark 1. In the special case when 
 
 r — mn = 0, 
 
 the centre («', i') of the curve will be at infinity, and the 
 theorem will not be directly applicable. 
 
 Remark 2. Since in the equation (4) the origin is at the 
 centre, this equation is that of the general curve of the second 
 degree referred to its centre as the origin. 
 
 190. Change of Direction of the Axes of Co-ordinates. 
 The next simplification of the equation will consist in remov- 
 ing the term in xy. To do this, lot us refer the curve to the 
 same origin as in (4), namely, the centre, but to a new system 
 of axes making an angle d with those of the original system. 
 This we do by the substitution (§ 27) 
 
 X = x' CO?, S — y' sin S\ 
 y z= x' sin ^ -\- y' cos S. 
 
 Making this substitution, the equation becomes 
 {m co^^d -\- n sin'd + 2Z %md cos d)x'^ 
 -\-{m sin'cJ -}- n cos^(J — 2Zsin 6 cos cJ),?/'^ 
 + [ (?i - m) sin 2d + 2? cos ^d^x'if = d\ 
 
 where we put — d' = a'p + b'q -\- d. (5) 
 
204 PLANE ANALYTIC GEOMETRY. 
 
 Substituting for the powers and products of sines and co- 
 sines their values, namely, 
 
 cos'(^ = ^(1 + cos 26^), 
 sin'(J = i(l - cos 2^), 
 2 sin S coe6 = sin 2^, 
 
 and then putting, for brevity, 
 
 li = {n — m)cos 2(y — 21 sin 2(5", ) / n 
 
 1c=\n- m)sin %6 + 2Z cos 26,) ^ ^ 
 
 this equation reduces to 
 
 (m -\-n- h)x" +(m + ^i + A)^/'' + 2kx'y' = 2d\ (6) 
 
 To make the term in a;'^' disappear, we must so determine 
 the value of 6 that ^ = 0. This gives 
 
 tan 2d = , 
 
 m — n 
 
 which determines the values of d. 
 
 Then from (c) we have, when k = 0, 
 
 h sin 2(J — y^ cos 2d = - 2Z =h sin 2d; 
 7i cos 2S -\- Jc sin 2d = ?i — w = A cos 2d. 
 
 The values of h and d are therefore given by the equations 
 
 h sin 2d = - 
 h cos 2d = 71 
 whence 
 
 sin 2d = - 2?; ) .^. 
 
 7i = >/(?w - ny + 4/^ (8) 
 
 Omitting accents, the equation (G) of the curve now re- 
 auces to 
 
 m- \-7i- V{7n-nY-\-4:r , , m + y^ + ^{^i - nY-jT r „ 
 
 2d' '^"^ 2d' r-i-(9) 
 
 The coefficients of x' and?/'* in this equation are always real, 
 but may be either positive or negative according to the sign of 
 d' and the values of m, n and I, 
 
OENERAL EQUATION OF THE SECOND DEGREE. 205 
 If, then, wc put 
 
 Id' 
 
 m + ?i - V{m - ny + 4.1' 
 m + n+ ♦/(?« - n)' + il' 
 
 (10) 
 
 the algebraic signs being so taken that a"" and h' without 
 sign shall be positive, the equation (9) still further reduces to 
 
 ±5±i;=i, (11) 
 
 which still represents the same locus as (1), only referred to 
 different axes and a different origin. 
 
 191. There are now three cases to be considered* 
 Case I. The algebraic signs in the first member both 
 
 negative. 
 
 Case II. The algebraic signs both positive. 
 
 Case III. The one sign positive and the other negative. 
 
 In the first case the equation is impossible Avith any real 
 values of x and y, because the first member will then be neces- 
 sarily negative, while the second is positive. The curve is 
 therefore wholly imaginary. 
 
 In the second case the equation is that of an ellipse whose 
 semi-axes are a and h. 
 
 In the third case the equation is that of an hyperbola whose 
 semi-axes are a and b. 
 
 We therefore conclude: 
 
 Theorem III. The locus of the eqiiation of the second de- 
 gree between rectangular co-ordinates is a conic section. 
 
 192. Special Kinds of Co7iic Sections. In order that the 
 equation (9) shall represent an ellipse we must have, by Case II., 
 
 m -[-n> V\m — nf + 4^. 
 Hence {m + nfy (m - ny + 4:P 
 
 and mn > 1% 
 
 or nm — V positive. 
 
206 PLANE ANALYTIC GEOMETRY. 
 
 Hence: 
 
 Theorem IV. The criterioii whether the general equation 
 (1) shall represent an ellipse or an hyperbola is given hy com- 
 paring the square of the coefficient of xy ivith four times the 
 product of the coefficieiits of x^ and y^. 
 
 If the square is algebraically less than four tiines the product, 
 the curve is an ellipse ; if greater, it is an hyperbola. 
 
 In special cases the equation may represent other lines than 
 the ellipse or hyperbola. We have, in fact, tacitly assumed 
 that the expressions a' and ¥ in (10) are both finite and de- 
 terminate. We have now to consider the case when either of 
 them is zero, infinity or indeterminate. 
 
 193, The Parabola. If in the equation (1) mn = r, 
 the preceding criterion will give neither a genuine ellipse nor 
 hyperbola, but a limiting curve between the two. We know 
 the parabola to be such a curve. In this case, also, the co-ordi- 
 nates a' and b' of the centre of the curve in (3) will be infinite, 
 so that the equation cannot be reduced to the form (4). But 
 when the centre of an ellipse or hyperbola recedes to infinity, 
 we know from Elementary Geometry that the curve becomes 
 a parabola. We shall now prove this result analytically. 
 
 Reduction in the Case of a Parabola, We have to con- 
 sider the special form of the general equation (1) in which 
 I = Vmn. The equation may then be written in the form 
 
 {nfx + n^yY -f ^x -^^y^d = 0. (12) 
 
 That is, in this case the sum of the three terms of the second 
 
 1 1 
 
 order forms the square of the linear expression m^x -j- n^y. 
 
 AVe may infer that the line whose equation is 
 nfx -j- n^y = 
 
 stands in some special relation to the curve. We shall there- 
 fore so change the direction of the axes of co-ordinates that 
 this line shall be the new axis of X. Taking the general 
 equation for this transformation, 
 
 ic = cc' cos d — y' sin d, \ / x 
 
 y = x' sin S -\- y' cos d, \ 
 
OENEEAL EQUATION OF THE SECOND DEGREE. 207 
 
 we see that they give 
 
 m X + 11^ y = {m^ cos S -\- tf sin S)x' 
 
 -\-(— i)v sin d -{- w cos ^)y'. {b) 
 
 In order, now, that the line 771 x -j- 7iy = may be identi- 
 cal witli the line y' = (which is the axis of X'), the coeffi- 
 cient of a;' in the above equation must vanish. That is, we 
 must have 
 
 ??i cos d + n^sin d = 0, (c) 
 
 or tan d = ;-; 
 
 , . „ tan S mi 
 
 whence sin o 
 
 Vl -f tan'd Vm + n 
 
 cos o — = -4- 
 
 Vl -\- tan^^d V7n + 7i' 
 
 — wi* sin (5' -|- ?t cos d = (m + ^)^ (^) 
 
 (5) and (a) now become 
 
 m X -\- n^y = \/{m -|- ?^)?/'; 
 nix' -\- 7nh/' n^y' — m^x' 
 
 By substitution, the equation (12) now becomes 
 
 and putting, for brevity, 
 
 (13) 
 
 1 p 
 
 _ qmi — pni 
 
 X -t 
 
 ~ {711 + 7l)l ' 
 
 iQ 
 
 _p7ni -\- q7ii 
 ~ (771 + ?^)§ ' 
 
 D- 
 
 d 
 
 771 -\- 7l' 
 
 the equation reduces to 
 
 7/' + Qy' - Fx' + D = 0. 
 
208 PLANE ANALYTIC GEOMETRY. 
 
 This, again, can be expressed in the form 
 
 (y' + kQY = W + P'-^' - D. (15) 
 
 We can still further simplify this equation by changing 
 the origin to the point whose co-ordinates are — ^Q and 
 
 ^^ . If the new co-ordinates referred to this origin are 
 
 X and y, we have 
 
 x = x' -\ -p ; 
 
 2/ = / + iS. 
 Then, by substitution, the equation becomes 
 
 y' = Px, (16) 
 
 which is the equation of a parabola whose parameter is j/P 
 referred to its vertex and principal axis. ^ 
 
 We therefore conclude: 
 
 Theorem V. The general equation of the second degree 
 represents a imralola luhen the square of the coef[icient of xij 
 is equal to four times the product of the coefficient ofx^ and if. 
 
 194. Case tvhen the Parameter is Zero. There is still a 
 special case of the parabola to be considered, namely, that in 
 which P = 0. From (16) it would then follow that y = for 
 all values of x. But this conclusion would be premature, 
 because the transformation (15) would then involve the plac- 
 ing of a new origin at infinity. We must therefore go back 
 to the equation (14), which, when P = 0, gives 
 
 y=-iQ± ViQ' - D; 
 
 that is, y may have either of two constant values. 
 
 Hence, when P = 0, the equation represents a pair of 
 straight lines parallel to the axis of Xand distant V^Q'^ — D 
 on each side of it. 
 
 195. General Case of a Pair of Straight Lines. 
 On reducing to the form (4), the absolute term d^ may 
 vanish, The reduction to the form (9) will then be impossi- 
 
GENERAL EQUATION OF THE SECOND DEGREE. 209 
 
 ble, because the coefficients of x^ and y^ will become infinite. 
 In this case, however, the equation (4) will bo 
 
 mx^ + 2lxij + ny"" = 0. (17) 
 
 If we factor this quadratic equation by any of the metliods 
 exphiined in Algebra, we may reduce it to the form 
 
 {ny + (^ + yl' - 'inn)x] X \ny -\- {I - VP - m?i)x} = 0, 
 
 or we may prove this equation by executing the indicated 
 multiplications and thus reducing it to the form (17). 
 
 Now this equation may be satisfied by equating either of 
 its two factors to zero. If we distinguish the values of y in 
 the two factors by subscript indices, we may have either 
 
 or 
 
 ^ _ - ^ + ^/^ 
 
 — mn 
 
 ^^ - 2n 
 
 
 -l-Vl' 
 2/. = 
 
 — mn 
 
 X 
 
 (18) 
 
 that is, to each value of x will correspond these two values of 
 y. But each equation (18) is that of a straight line passing 
 through the origin. We therefore conclude: 
 
 Theorem VI. When, on reducing the general equation of 
 the second degree to the centre, the absolute term vanishes, 
 the equation represents a ])air of straight lines. 
 
 If we have Z* < mn, the lines will both be imaginary. 
 But in this case there will be one pair of real values of the co- 
 ordinates, namely, a; = and y = 0. Hence, 
 
 If, in the case supposed in the preceding theorem, the lines 
 lecome imaginary, the equation can he satisfied ly only a single 
 real 'point. 
 
 This result is also evident by a comparison of equations 
 (9), (10) and (11), because when d' = and P < 7nn, we 
 have an ellipse of which both the axes are zero, and this can 
 be nothing but a point. 
 
 On the other hand, if both the axes of an hyperbola be- 
 come zero, it reduces to a pair of straight lines. 
 
 We' have thus found two seemingly distinct cases in 
 
210 PLANE ANALYTIC GEOMETRY. 
 
 which the conic is reduced to two straight lines: the one when 
 r — 7nn = und P = 0; the other when d' = 0. We shall 
 now show that the former cases may bo combined with the 
 latter. 
 
 If in the expression (5) for d^ we substitute for a' and b' 
 their values (3), it will become 
 
 — d' = i^(^^i^ ~ ^g) + g(^"g^ ~ ^P) + ^(^' ~ ^^^^0 
 r — 7nn 
 
 Now let us put 
 
 R =p{np — Iq) -f- q{mq — Ip) -\- d(r — m7i), (19) 
 
 so that we have 
 
 R = d'{imi - r). (20) 
 
 If we square the value (13) of P and note that we are 
 considering the case when mn = V, we have 
 f {m + nyP' = mq' - %lpq + nf 
 
 ^pi^wp - Iq) + q{mq - Ip). 
 This expression is zero, by hypothesis, since P = 0. Com- 
 paring it with (19) and noting thatZ^ -- nm = 0, we see that 
 the value of R vanishes in this case as it does when d' = 0. 
 We therefore conclude that i2 = is the condition that the 
 conic shall reduce to a pair of straight lines. 
 
 196. Summary of Coiichisions. The various conclusions 
 which we have reached may be recapitulated as follows: 
 The general equation of the second degree, 
 
 rax" + 2lxy + ny"" + ^x -\- 2qy -\r d - 0, 
 
 represents 
 
 An ellipse when r < mu] 
 A parabola lohen T = mn\ 
 An hyperbola when V > mn. 
 Also, in spfecial cases, 
 
 The ellipse may be reduced to a point; 
 The parabola to a pair of parallel straight lines; 
 The hyperbola to a pair of intersecting straight lines. 
 But since, in the first case, the point- ellipse is defined as 
 the real intersection of a pair of imaginary straight lines, we 
 
GENERAL EQUATION OP THE SECOND DEGREE. 211 
 
 may describe all three of these cases as one in which the conic 
 is reduced to a pair of straight lines, and sum up the conclu- 
 sion thus: 
 
 If the coefficients in the gc7ieral equation of a conic satisfy 
 the condition 
 
 p{np - Iq) + q(mq — Ijj) + d{r - mn) = 0, (19) 
 
 the conic will he reduced to a pair of straight lines. If we 
 have 
 
 r < m7i, the lines are imaginary; 
 
 r = 7n7i, the lines are real, and parallel or coincide7it; 
 
 r > mn, the lines are real and intersecting. 
 
 19*7. All these forms are conic sections. That the 
 ellipse, parabola and hyperbola are such sections is shown in 
 Geometry. 
 
 When the cutting plane passes through the vertex of the 
 cone, the section is a point or a pair of intersecting straight 
 lines according to the position of the plane. 
 
212 PLANE ANALYTIC OEOMETRY. 
 
 When the vertex of the cone recedes to infinity, the base 
 remaining finite, the cone becomes a cylinder, and the sec- 
 tion parallel to the elements is a pair of parallel straight lines. 
 
 Remark. A conic section is, for brevity, frequently 
 called, a conic simply, and we therefore designate all loci of 
 the second degree as conies. 
 
 198, Similar Conies. From equation (10) it follows 
 that the ratio a : b of the semi-axes depends only upon the 
 coefficients m, n and I of the terms of the second order in 
 the general equation. Since we have, for the eccentricity, 
 
 a 
 
 it follows that the eccentricity depends only on the same co- 
 efficients, I, m and n. 
 
 Moreover, the angle d which the principal axes of the 
 conic form with the original axes of co ordinates depends only 
 on these same coefficients. Hence, using the definitions. 
 
 Similar conies are those which have the same eccen- 
 tricity or (which is the same thing) the same ratio of the 
 two principal axes; Similar conies are said to be similarly 
 placed when their corresponding axes are parallel, — we have 
 the theorem: 
 
 Theorem VII. All conies whose equations have the same 
 terms of the second degree vn the co-ordinates are similar 
 and similarly placed. 
 
 199. Theorem VIII. A conic section may he made to 
 pass through any five points in a plane. 
 
 Let us divide the general equation (1) by d, and, distin- 
 guishing the new coefficients by accents, we have 
 
 m'x' + n'y"" + Uxy + 2p'x + ^'y + 1 = 0. {a) 
 
 Now if (a;,, ?/,), (ic,, ?/J, (2^3, 2/3), (x^, y^), {x„ y,) are the 
 five given points in the plane, we have, by substituting in the 
 last equation the co-ordinates of these five points for the gen- 
 eral co-ordinates x and y, the five following equations of con- 
 dition: 
 
GENERAL EQUATION OF THE SECOND DEGREE. 213 
 
 m\' + oi'y^' + ^ZVx,y^ + 2p% + 2q'y, + 1 = 0; 
 m'x,' + n>,^ + 2Z'.r,y, + 2/;'a;, + 2q'y, + 1 = 0; 
 m^x: + n'i/3» + 2/^2/3 + ^i^'^3 + ^'y. + 1 = 0; 
 m^x: + 7iV,» + n'x,y, + 2i>X + ^'y. + 1 = 0; 
 7;iV + ^i'l/,^ + %l\y, + 2;;':^:, + 2q'y, + 1 = 0; 
 
 from which the coefficients m', n\ V, 'p' and q' may be found, 
 since x^, ;/„ a*^, y^, etc., are hnoioii quantities. Substituting 
 these values of m', n' , etc., in the general equation (a), the 
 resulting equation of the second degree in x and y will be 
 that of the required conic section. 
 
 Cor, Since the equations of condition are all of the first 
 degree with respect to m* , n' , V, y' and q' , each of these 
 quantities has only one value; tlierefore only one conic section 
 can be passed tJiroiigh five given points on a plane. 
 
 Example. Let it be required to pass a conic section 
 through the five points (2, 1), (- 1, - 3), (0, 3), (1, 0), 
 (3, - 2). 
 
 The equations of condition which determine the coefficients m, n, ?, 
 etc., are (omitting the accents) 
 
 4m-{- n-\- 4l-\-4p + 
 
 m -}- 9/i + 6^ — 2j) — 
 
 9/1 + 
 
 Qjn + 4n — 121 + 6p - 
 
 2? + 1 = 0; 
 6g + l = 0; 
 6g -f 1 = 0; 
 + 1=0; 
 iq + 1 = 0; 
 
 
 >m which we find 
 
 
 
 33 41 1 
 ""64' ^ 192' 32"' 
 
 97 
 
 ^ - 128 ' 
 
 59 
 ^=384 
 
 Substituting these values in the general equation 
 
 mx'^ + n7f + 2lxy + 2px + 2gy + 1 = 0, 
 and clearing of fractions, we have 
 
 99.?^2 _ 41^3 _ i2xy - 291a; + 59y + 192 = 0, 
 which is the equation of an hyperbola, since I- — mn is a positive quantity. 
 
 If one of the given points should be the origin, the corresponding 
 equation would be the impossible one 1 = 0. In this case we should 
 have to divide by some other coefficient than d. 
 
214 PLANE ANALYTIC OEOMETRT. 
 
 300. Intersection and Tangency of Conies. 
 Theorem IX. Tico co7iics in general intersect each 
 
 other in four points. 
 
 Proof. The co-ordinates x and y of the points of inter- 
 section of two conies are given by the roots of two equations 
 of the second degree in x and y. Now, it is shown in Algebra 
 that when we eliminate an unknown quantity from two quad- 
 ratic equations, the resulting equation in the other unknown 
 quantity will, in general, be of the fourth degree. This equa- 
 tion will therefore have four roots, thus giving rise to four sets 
 of co-ordinates of the points of intersection. 
 
 Remark. The roots may be all four real; one pair real 
 and one pair imaginary; or all four imaginary; and, in any 
 case, the two roots of a pair may be equal. 
 
 According as this happens the conies are said to intersect 
 in real, imaginary or coincident points. In the latter case 
 they are said to touch each other at the coincident points. 
 
 Cor. Tivo conies may touch each other at two jwints and 
 no more. 
 
 301. Families of Conies. Let us put, for brevity, 
 P' = mV + 2Vxy + n'y'' + 2fx + ^'y + d' ; 
 
 etc. etc. etc. ; 
 
 that is, let us represent by P', P", etc., any functions of the 
 second degree in the co-ordinates. 
 
 Theorem X. If P' = and P" = are the equations 
 of any two different conies, then the equation 
 
 }xP' + ;iP" = (20) 
 
 {where }i and \ are constants) tvill represent a third conic 
 passing through the four points of intersection of the other 
 tivo. 
 
 For, first, we see by substitution of the values of P' and 
 P" that the equation (20) is of the second degree in the co- 
 ordinates. Hence its locus is some conic. 
 
 Secondly, every pair of values of x and y which make 
 
GENERAL EQUATION OF TUE SECOND DEGREE. 215 
 
 both P' = and P" = must also satisfy the equation 
 )uP' 4- A.P" = 0. Hence every point common to P' and P" 
 must belong to the locus of (20); that is, this locus passes 
 through all the points, real and imaginary, in which i^' and 
 P" intersect. The number of these points is four. 
 
 By giving different values to the ratio X : //, any number 
 of conies passing through the same four points may be found. 
 We ma}^, without loss of generality, suppose // = 1 in this 
 theory, because the locus (20) depends only on the ratio X : jn 
 
 Def. A system of conies all of which pass through the 
 same four points is called a family of conies. 
 
 203. Theorem XL In a family of conies tiuo and 
 no more are parabolas. 
 
 Proof If, in the expression 
 
 P =: P' -\- AP", 
 
 we substitute for P' and P" their values, we shall have, in P, 
 
 Coefficient of x^ = m' -\- Xm" = m; 
 Coefficient of ^ = 7i' -f '^^^^ = ''^ 
 Coefficient of 2xy = V ^ XI" = I 
 
 The condition that the curve P = ishall be a parabola 
 then becomes 
 
 = r — mn 
 = (l"^ - m!'n*')V-\- {%VV'- m'n"- m!'n')X + Z" - mV. 
 
 This is a quadratic equation in X, which therefore gives 
 two values of A, and thus two expressions for P, each of which, 
 equated to zero, is the equation of a parabola. Q. E. D. 
 
 303. Theorem XII. In a family of co7iics three, and 
 no more, may he pairs of lines. 
 
 Proof. Forming the expression P' + XP'\ we find the 
 coefficients of the general equation to become 
 
 m = m' -\- Am"; 
 
 n = n' -f Xn"\ 
 
 etc. etc. 
 
216 
 
 PLANE ANALYTIC GEOMETRY. 
 
 In order that a conic of the family may be a pair of lines 
 it is necessary and sufficient that its coefficients satisfy the 
 condition (19). Each term of (19) is of the third degree in 
 the coefficients. Ilence the entire condition gives an equation 
 of the third degree in A, which has three roots. Ilence we 
 have three expressions of the form F' + \P", each of which, 
 when equated to zero, gives a pair of lines. Q. E. D. 
 
 Remark. If we call A, B, C and D the four points of 
 intersection of the family, the three pairs of lines which be- 
 long to it will pass as follows: 
 
 One pair through AB and CD respectively; 
 One pair through ^(7 and BD respectively; 
 One pair through AD and ^(7 respectively; 
 
 and the three pairs will form the sides and diagonals of a 
 quadrilateral. 
 
 204. Theorem XIII. If loe tahe any point {x^, y^) 
 at pleasure in the plane of a family of conies, then one conic 
 of the family, and no more, will pass through this point. 
 
 Proof Since the equation 
 
 must be satisfied for the value {x^, y^) of the co-ordinates 
 X and y which enter into it, we have 
 
 (m' + Xm'')x^' + {n' + 'Xn")y,' + 2(^' + W)x,y, + etc. = 0. 
 
GENERAL EQUATION OF THE SECOND DEGREE. 217 
 
 Since x^, ?/, and all the symbols except A. in this equation 
 are known, it is an equation of the first degree in X and 
 so has but one root. This root may be expressed in the form 
 
 in which P/ and P/' represent the values of P' and P" when 
 x^ and y^ are substituted for x and y. There being but one 
 value of A, only one conic of the family can pass through 
 the point {x^, ?/,). 
 
 205. THEOREii XIV. The equation 
 
 P' 4- \p" = (a) 
 
 may, by giving all real values to A, represent every possible 
 
 conic passing through the four i^iter sections of P' and P" , 
 
 For, let G be any conic passing through the four points. 
 
 P ' 
 Take any fifth point (x^, y^ on (7, and put A, = — -^^. The 
 
 equation (a) will then be satisfied identically when in it we put 
 
 x-x^, y = y^, 
 
 because it will become 
 
 P ' 
 P ' — ^ P " = 
 
 Hence, with this value of A, (a) will represent a conic of the 
 family passing through the point (x^, y^. But only one 
 conic can pass through five points. Hence the conic thus 
 found will be C. 
 
 306. Relation of Focus and Directrix to the General 
 Equation. Let P^Cbe any conic section; GX, OY, rectan- 
 gular axes. Let AQ, the axis of the curve, make an angle 
 AGX= a with the axis GJC. And let (x, y) be the co-ordi- 
 nates of any point P; {h, h), the co-ordinates of the focus F\ 
 and r = GD, the distance from the origin to the directrix 
 DK, Join PP, and draw PE perpendicular to DK, and 
 
218 
 
 PLANE ANALYTIC GEOMETRY. 
 
 FH i")arallel to OX. Then, by the definition of a conic section 
 given in § 181, Chapter VIL, we have 
 
 PF 
 
 p-^ = e, the eccentricity, 
 
 and therefore PF = ePE 
 
 and PE = x cos a -\- y ^m a — r. 
 
 Hence PF = e{x gos a -\- y sin a — r), 
 
 PF' = FH' + PH\ 
 or (x cos a -{- y ^m a — rye' = (x — h)' -\- (y — Icf. 
 
 "Yl 
 
 Expanding and collecting terms, we have 
 
 (1 — e' cos'^a:)^;'^ + (1 — e' dn'a)y' — 2e' sin a cos a xy 
 -\- (2e'r cos a — 2h)x + (J^e'i' sin a — 2k)y 
 + {h' + F - e'r') = 0. (21) 
 
 To compare this with the general equation, we must divide 
 both it and the general equation by their absolute terms, in 
 order that the two may have the same coefificients. Supposing 
 
 171 
 
 the general equation thus divided, and writing 7n for -r, n for 
 
 -7, etc. : also putting, for brevity, 
 
 A = cos a; fx = sin a; 
 
GENERAL EQUATION OF THE SECOND DEGREE. 219 
 
 we find, by comparing coefficients of corresponding terms in 
 the two equations, 
 
 1 - e'X' 
 
 or 
 
 A' + F - eV ~ ' 
 
 
 
 1 -c'A»=m(F+;t' 
 
 -cV); 
 
 {a) 
 
 1 -e'tx' = n{k-' + V 
 
 - eV); 
 
 (i) 
 
 - e'XM = l{k' + V 
 
 - e'r'); 
 
 (c) 
 
 A'Ar -h =;j(F + /i" 
 
 - «V); 
 
 id) 
 
 e'nr -h^ g{l:' + h' 
 
 - eV). 
 
 ie) 
 
 These five equations completely determine the five quanti- 
 ties a, r, h, h and e, and hence the focus, directrix and 
 eccentricity of the conic, in terms of the coefficients of the 
 general equation. 
 
 EXERCISES. 
 
 1. Investigate the locus represented by the equation 
 
 g 
 Here we have 7n = 4; 7i = l; I = —. 
 
 z 
 
 Then rnn- I'' = 4: - ^ = -\-\', 
 
 4 4 
 
 therefore the locus is an ellipse. 
 
 2. Find the co-ordinates of the centre of the conic repre- 
 sented by 
 
 bx" + ^' + 2xy - Zlx - 2y -\- 100 = 0, 
 and find the angle between the axis of the curve and the axis 
 of X. 
 
 3. "What curve does y^ = 3{xy — 2) represent? 
 
 4. Determine the locus y^ = d{x — 7) and the angle its 
 axis makes with the axis of X. 
 
 5. Determine the locus of x^-\- y^— 6xy — 6x-\-2y -\-6 = 0. 
 Find co-ordinates of the centre, and the angle the axis of the 
 curve makes with the axis of X. Aiis. (0, — 1); 135°. 
 
 6. If Ay"" + Bxy -\- Cx' + Dy + Bx -{- F = be the 
 equation of a conic section, show that 
 
 Bx 4- 2Ay -{- D = 
 is the equation of a diameter of the locus. 
 
220 PLANE ANALYTIC GEOMETRY. 
 
 7. From the equation (9) find the two conditions that the 
 equation of the second degree shall represent a circle. 
 
 8. Find in the same way the two conditions that the gen- 
 eral equation shall represent an equilateral hyperbola. 
 
 9. What locus is represented by the equation 
 
 li'x'' -\- mxy -\- k^'y' — c% 
 
 when m = lih'^ 
 
 10. Find the semi-parameter of the parabola 
 
 {x - yY = ax. 
 
 11. What angle do the asymptotes of the hyperbola 
 
 mx^ — xy =^ a 
 
 make with the transverse axis? 
 
 12. If we have the two conies 
 
 mx"" -f 2lxy + ny'' + ^px -^ 2qy -^ d = 0, 
 mx^ + ^Ixy -\- ny"" — 2px — 2qy -{- d = 0, 
 
 show that the line joining their centres is bisected by the 
 origin. 
 
 13. The co-ordinates x and ?/ of a moving point are ex- 
 pressed in terms of the time t by the equations 
 
 X = mt -}- a; y = mt -\- b. 
 
 What is the equation of the line described by the point? 
 
 14. If the co-ordinates are given by the equations 
 
 X = mt, y = nt', 
 
 show that the curve is a parabola, and express its parameter. 
 
 15. What condition must the coefficients of the general 
 equation (1) of the second degree satisfy that the curve may 
 pass through the origin of co-ordinates? 
 
 16. Write the equation of that conic formed of a pair of 
 straight lines through the origin whose slopes are m and — m. 
 
 17. Do the same thing when the lines are to intersect in 
 the point (a, h). 
 
 18. What is the condition that the principal axes of a 
 conic shall be parallel to the axes of co-ordinates? (See §190. ) 
 
GENERAL EQUATION OF THE SECOND DEGREE. 221 
 
 19. Express the points in which the locus of the equation 
 
 x" - 2:c?/ + ?/' + 3a; - 4 = 
 
 cuts the respective axes of co-ordinates. 
 
 Ans, The axis of X, (1, 0) and (- 4, 0); 
 The axis of Y, (0, 2), (0, - 2). 
 
 20. What condition must the coefficients of (1) satisfy that 
 the curve may be tangent to the axis of X and to the axis of 
 Irrespectively? 
 
 Ans. p^ = md for the axis of X; 
 q^ = nd for the axis of Y, 
 
 The solution is very simple, if it is remembered that the curve is to 
 cut the axis in two coincident points. 
 
 21. Find the equation of that conic which cuts the axis 
 of X at points whose abscissas are — 2 and + 4, the axis of 
 l^at points whose ordinates are — 1 and + 2, and whose princi- 
 pal axes are parallel to the axes of co-ordinates. 
 
 Ans. x" + 4:y'^ — 2a; — 4?/ — 8 = 0. 
 
 22. Show that in the general equation (1) the line 
 
 mx -}- ly — p = 
 
 bisects all chords parallel to the axis of X. Find also the line 
 which bisects all chords parallel to the axis of Y. 
 
 Begin by solving the general equation as a quadratic in x so as to ex- 
 press X in terms of p, and vice versa. 
 
 23. How many points are necessary to determine a para- 
 bola? An equilateral hyperbola? 
 
 24. Mark five points at pleasure on a piece of paper. 
 Can you find any criterion for distinguishing at sight the fol- 
 lowing cases? — 
 
 I. The five points lie on one branch of a conic (ellipse or 
 hyperbola). 
 
 II. The conic is an hyperbola having three of the points 
 on one branch and two on the other. 
 
 III. It is an hyperbola having four points on one brancli 
 and one on the other. 
 
 Suppose a string drawn tightly around all the points, and 
 note the number of points the string will not reach. 
 
222 PLANE ANALYTIC GEOMETRY. 
 
 25. Find the equation of a parabola which shall touch the 
 axis of X at the point whose abscissa is + ^> ^^^ the axis of 
 JTat the point whose ordinate is + 1- 
 
 Alls, x^ — 4:xy + ^y^ — 4a; — 8y -{- 4, = 0. 
 
 26. The base of a triangle has a fixed length, and the 
 escribed circle below this base is required to touch it at a 
 fixed point. Find the locus of the point of intersection of the 
 two sides of the triangle. 
 
 27. A line passes through the fixed point (0, i) on the 
 axis of Y and intersects the axis of X and the fixed line 
 y = mx. Find the locus of the middle point of the segment 
 of the line contained between the fixed line and the axis of X. 
 
 28. Investigate the locus of the point the differences of 
 the squares of whose distances from the axis of X and from 
 the line y = mx is the constant quantity ^^ 
 
 29. The base <^ of a triangle and the sum of the angles at 
 the two ends of the base are both constant. Investigate the 
 locus of its vertex. 
 
 30. Each abscissa of the circle x"^ -\- y^ = r^ is increased by 
 m times its ordinate. Find the locus of the ends of the lines 
 thus formed. 
 
 31. Investigate the locus of the middle points of all chords 
 of an ellipse which pass through a fixed point. 
 
 32. The circle x^ + 2/' = ^* has two tangents intersecting 
 in a movable point P and cutting out a fixed length a from 
 a third tangent y = r. Investigate the locus of P. 
 
 33. Show that the equation of that pair of straight lines 
 formed of the axes of co-ordinates is xy = 0. 
 
PART II. 
 GEOMETRY OF THREE DIMENSIONS. 
 
 CHAPTER I. 
 
 POSITION AND DIRECTION IN SPACE. 
 
 207. Directions and Angles in Space. Two straight lines 
 cannot form an angle, as that term is defined in elementary 
 geometry, unless they intersect. Two lines in space will, in 
 general, pass each other witliout intersecting. Hence we 
 cannot speak of the angle between such lines unless we extend 
 the meaning of the word angle. Now the following theorem 
 is known from solid geometry: 
 
 If ive have given any two lines, a and h, in space; 
 
 and if ive take any point P at pleasure; 
 
 and if through P loe draw tioo lines, PA arid PB, par- 
 allel to a and h respectively, — 
 
 then, so lo7ig as we leave a and h unchanged, the angle AP B 
 will have the same value no matter where ive take the point P. 
 
 "We therefore take this angle as the measure of the angle 
 between the lines a and h. This measure may be considered 
 as expressing the difference of direction between the lines 
 a and h, and the word angle, as applied to two non-intersecting 
 lines, will be understood to mean their difference of direction. 
 We thus have the following definition and corollary: 
 
 Def Tlie angle between two non-intersecting lines is 
 measured by the angle between any two intersecting lines 
 parallel to them. 
 
 Cor. If we have two systems of parallel lines in space, the 
 07ie a, a* , «", etc., the other h, h', h", etc., then the angles hctiveen 
 any line of a and any line of b will all be equal to each other. 
 
224 GEOMETRY OF THREE DIMENSIONS. 
 
 208. Projections of Lines. The projection of a finite 
 line PQ upon an indefinite line Xis the lengtli oi X inter- 
 cei)ted by the perpendiculars dropped upon it from the two 
 ends of PQ. 
 
 Theorem I. The projection of a line is eqnal to the 2Jro- 
 duct of its length by the cosine of the angle ichich it forms 
 with the line on lohich it is projected. 
 
 To prove the theorem, pass through each of the termini, 
 P aud Q, a plane perpendicular to JT.* The planes will be 
 parallel to each other and will cut X at the termini of the 
 projection of PQ, which we may call P' and Q'. Through 
 P' draw P'Q'^ \\ PQ and intersecting the plane through Q 
 in Q". We shall then have 
 
 P'e" = PQ', 
 
 (being parallels between parallel planes;) 
 
 P'Q' ^ P'Q" cos Q"P'Q' 
 
 = PQ cos (angle between P'Q' and PQ). Q.E.D. 
 
 Remark. By assigning a positive and negative direction 
 to the two lines, the algebraic sign of the projection will be 
 determined. It will be positive or negative according as the 
 angle between the positive directions of the two lines is less 
 or greater than a right angle. The following theorem is a 
 result of this convention, combined with the principles of 
 Trigonometry: 
 
 Theorem II. If we have any broken line in space, made 
 up of the consecutive straight lines AB, BG, CD, etc., . . . GH, 
 ■which lines form the angles a, ft, y, etc., tvith the line of i^ro- 
 jection X; 
 
 and if we project this line upon X by dropping perpendicu- 
 lars A A\BB', GG', etc., . . . . HH', meeting X at the points 
 A', B', C', D', etc., .... H'— 
 
 then the length A'FP zvill be the algebraic sum of the 
 
 * No figure is drawn for this demonstration, because two non-inter- 
 secting lines in space cannot be represented on paper. If the student 
 cannot readily conceive the relation, he should take two rods or pencils 
 to represent the lines. 
 
POSITION AND DIRECTION IN SPACE. 
 
 225 
 
 separate lengths A'B\ B' C , CD' , etc., these separate lengths 
 being considered j^ositive when taken m one direction, 7iegative 
 when taken in the oppodte direction, and will be expressed 
 by the equation 
 
 A'H' = AB cos a -{- BC cos /3-\- CD cos y + etc. 
 
 209. Co-ordinate Axes and Flaiies in Space. The position 
 of a point ill space maybe defined by its relation to three straight 
 lines intersecting- in the same point and not lying in a plane. 
 
 Three such lines of reference are called a system of co- 
 ordinate axes in space. 
 
 The point in which the axes intersect is called the origin 
 of co-ordinates, or simply the origin. 
 
 The three axes are designated by the letters JT, Y and Z 
 respectively. 
 
 The co-ordinate axes, taken two and two, lie in three planes, 
 one containing the two axes X and Y, another l^and Z, a 
 third Z and X. 
 
 These planes are called co-ordinate planes. They 
 are distinguished as the plane of XY, the plane of YZ, and 
 the plane of ^X respectively. 
 
 The several angles which the axes of co-ordinates make 
 witli each other are arbitrary. But, for elementary purposes, 
 it is most convenient to suppose each to form a right angle with 
 
226 GEOMETRY OF THREE DIMENSIONS. 
 
 the other two. The following conclusions then result from 
 solid geometry: 
 
 I. Each axis is perpendicular to the plane of the other two. 
 
 II. Each plane is perpendicular to the other two planes. 
 
 III. Every line or plane perpendicular to one of the planes 
 is parallel to the axis which does not lie in that plane. 
 
 IV. Every line or plane perpendicular to one of the axes 
 is parallel to the plane of the other two. 
 
 V. If the centre of a sphere lies in the origin, the intersec- 
 tions of the co-ordinate axes and planes with its surface form the 
 vertices and sides of eight trirectangular spherical triangles. 
 
 VI. To each plane corresponds the axis perpendicular to 
 it, which is therefore called the axis of the plane. 
 
 210. Co-ordinates, The position of a point in space is 
 defined by its distances from the three co-ordinate planes of a 
 system, each distance being measured on a line parallel to the 
 axis of the plane. When the axes are rectangular, these direc- 
 tions will be perpendicular to the planes. The notation is: 
 X = distance from plane YZ] 
 y = distance from plane ZX; 
 z = distance from plane XT. 
 
 To distinguish between equal distances on the two sides 
 of a plane, distances on one side are considered positive, 
 on the other negative. 
 
 The positive direction from jeach plane is the positive di- 
 rection of the axis perpendicular to it. 
 
 It is, of course, a matter of convention which side we take 
 as positive and which negative. A certain relation between 
 the positive directions is, however, adopted in ph3^sics and 
 astronomy, and should be adhered to. It is this: 
 
 The positive side of the plane of XY is Y 
 that from lohich we must looh in order that 
 the axis OX would have to turn in a direc- 
 tion the opposite of that of the hands of a 
 watch in order to talce the positio7i OY. 
 
 If we conceive the plane of XY to be o' ^X 
 
 horizontal, the axis of Z will be vertical, and, supposing the 
 
POSITION AND DIRECTION IN SPACE. 227 
 
 axes of X and Y to be arranged as in plane geometry, the 
 positive side of tlie plane will be the upper one.* 
 
 The following propositions respecting certain relations of 
 signs of co-ordinates to position should be perfectly clear to 
 the student : 
 
 I. The co-ordinate planes divide the space surrounding the 
 origin into eight regions, distinguished by the distribution 
 of + and — signs among the co-ordinates. 
 
 Imagine the axis of X to go out positively toward the east; 
 Imagine the axis of I^to go out positively toward the north; 
 Imagine the axis of -^to go out positively upward, 
 
 and the point of reference to be the origin. Then — 
 
 II. For all points above the horizon z will be positive; for 
 all points below it, negative. 
 
 III. For all points east of the north and south line x will 
 be positive; for all points west of it, negative. 
 
 IV. For all points north of the east and west line y will be 
 positive; for all points south of it, negative. 
 
 311. How the Co-ordinates define Position. Let us first 
 suppose that the only information given us respecting the 
 position of a point P is its co-ordinate 
 
 X — a, (a) 
 
 a being a given quantity. 
 
 This is the same as saying that P is at a distance a from 
 the plane YZ. In order that a point may be at a distance a 
 from a plane, it is necessary and sufficient that it lie in a par- 
 allel plane, such tiiat the distance between the two planes is a. 
 
 Hence the proposition informs us that P lies in a certain 
 plane. 
 
 * The author regards it as unfortunate that many mathematical 
 writers, in treating of analytic geometry, reverse the arrangement of axes 
 in space universally adopted in astronomy and physics. Uniformity in 
 this respect is so desirable that he has not hesitated to adhere to the latter 
 arrangement. 
 
 It may be remarked that, in drawing figures, the axes are represented 
 as seen from different stand-points in different problems, the best point 
 of view for each individual problem being chosen. 
 
228 OE0METE7 OF THREE DIMENSIONS. 
 
 If we are informed that 
 
 y = ^, 
 then P must lie in a plane parallel to ZX, at the distance 1) 
 from it. 
 
 If both propositions, x = a and y = b, are true, then P 
 must lie in both planes. Hence it must lie on their line of 
 intersection, which line will be 
 
 parallel to the axis of Z, 
 parallel to the planes ZX and YZ, 
 and perpendicular to plane XY. 
 
 If it is also added that 
 
 2; = c, 
 
 the point P lies in a third plane, parallel to XY, Lying in 
 all three planes, its only position will be their common point 
 of intersection. 
 
 Hence: 
 
 Tlie position of a point is completely determined luhen its 
 three co-ordinates are given. 
 
 NoTATioi^. By point {a, h, c) we mean the point for 
 which X = a, y = h and z = c, 
 
 212. Paralleloinpedon formed hy the Co-ordinates. 
 
 Let be the origin; OX, OY, OZ, the axes; P, the point; 
 PR, PS and PQ, parallels to the axes terminating in the sev- 
 eral planes. Then, by definition, the co-ordinates of P will 
 
 be 
 
 x = EP = VS = OT = WQ; 
 
 y=: SP = VP= 0W= TQ; 
 
 z = QP = TS = OV = WR. 
 
 We shall then have, by considering the three planes which 
 
 contain these co-ordinates. 
 
 Plane EPS || plane XY; 
 Plane SPQ \\ plane YZ; 
 Plane QPR \\ plane ZX. 
 
 Hence the three planes which contain the co-ordinates, 
 together with those which contain the axes, form the six faces 
 of a parallelopipedon. This figure has 
 
POSITION AND DIRECTION IN SPACE. 
 
 229 
 
 Four edges = and 
 Four edges = and 
 Four edges = and 
 
 to co-ordinate x', 
 to co-ordinate y\ 
 to co-ordinate z. 
 
 X 
 
 213. Since there are four equal lines for each co-ordinate, 
 we may use any one of these four in constructing the co-ordinate. 
 Sometimes it is advantageous to choose such lines that, taking 
 the co-ordinates in some order, the end of each shall coincide 
 
 with the beginning of the next following, the end of the third 
 being the point. For example, we may take, in order, 
 
 X = OT; 
 y = TQ; 
 z = QF. 
 
230 GEOMETRY OF THREE DIMENSIONS. 
 
 The three co-ordinates will then form a series of three lines, 
 each at a right angle with the other two. 
 
 Again, each face of the parallelopipedon being perpendicu- 
 lar to four edges, it follows that the diagonal PT will be the 
 perpendicular from P upon the axis of X, and that the like 
 proposition will be true for the other axes. Hence 
 
 The rectangular co-ordinates of a point are equal to the 
 segments of the axes contained betioeen the origin and the per- 
 pendiculars dropped from the point upon the respective axes. 
 
 EXERCISES. 
 
 1. If from the point {a, h, c) we draw lines to the several 
 points {— a, b, c), (a, — b, c), {a, b, — c), (— a, — b, c), 
 (a, — b, — c), {— a, b, — c), {— a, — J, — c), define in what 
 seven points these lines will cut such of the co-ordinate 
 planes as they intersect. 
 
 2. If perpendiculars be dropped from a point {a, b, c) upon 
 the three co-ordinate axes, show that the lengths of the per- 
 pendiculars will be Vd' + b\ Vb' + c' and Vc' + a\ 
 
 3. If we take, on each axis, a point at the distance r from 
 the origin, what will be the mutual distances of the three 
 points from each other, and of what figure will they and the 
 origin form the vertices ? 
 
 ^sZt'-jL ^' ^^' ^^^ ^^^® ^^^^ ^^ ^' 1^ and Z respectively, we take 
 the points P, Q and R, and from the origin drop OLL QR, 
 i--^^-*-^MlRP and ONLPQ, show that 
 
 or ^ OM' ^ ON' OP' ^ OQ' ^ OR' 
 
 214. Problem. To find the distance of a point (x, y, z) 
 from the origin, and the a7igles ichich the line joining it to the 
 origin makes icith the co-ordinate axes. 
 Let P be the point, and let us put 
 
 r, the distance OP from the origin; 
 a, y5, r, the angles POX, POY and POZ 
 which the line makes with the axes. 
 Then— 
 I. Because OP is the diagonal of a rectangular parallelo- 
 
POSITION AND DIRECTION IN SPACE. 231 
 
 pipedon whose edges are PQ, PR and PSy we have, by 
 Geometry, 
 
 OP' = PQ' 4- PR' + PS'; 
 that is, 
 
 r' = x'-\- y' + z' (1) 
 
 and r = Vx' -j- y' ~\- z', 
 
 which gives the distance of the point from the origin. 
 
 II. Again, supposing the same construction as in § 212, we 
 have 
 
 pro = a right angle. 
 Hence 
 
 0T= OP COS POX, 
 
 or, from the equality of the parallel edges, 
 
 a: = r cos a. 
 In the same way i y x 
 
 y = r cos J3; « ^ ' 
 
 z = r cos y. 
 
 The required values of the cosines of the angles are, there- 
 fore, 
 
 X X ^ 
 
 cos a = — = — ; 
 
 r V,c^ -{-f-\-z^ 
 
 z z 
 
 cos ;/ = - = 
 
 ^ Vx"" + y'' + z"" 
 
 (3) 
 
 Theorem III. Tlie sum of the squares of the cosines of 
 the angles ichich a line through the origin makes with three 
 rectangular axes is unity. 
 
 Proof Adding the squares of the last set of equations, 
 we have 
 
 cos'a + cos'/? + oosV = J^—J = 1, (i) 
 
 which proves the theorem. 
 
 This theorem enables us to find any one of the angles a, 
 /? and y when the other two are given. 
 
232 
 
 GEOMETliT OF THREE DIMENSIONS. 
 
 215, Problem. To express the distance betioeen two 
 points given hy their co-ordinates, and the angles which the 
 line joining them forms loith the axes of co-ordinates. 
 
 Let P and F' be the points, and let 
 
 P have the co-ordinates x, y, z\ 
 
 P' have the co-ordinates x\ 
 
 y 
 
 I* 
 
 IP 
 
 ■>ivr 
 
 Through each of the points P and P' pass three planes 
 parallel to the co-ordinate planes. These planes will form the 
 faces of a rectangular parallelepiped of which the edges are 
 
 P'M = x' -x\ 
 P'N = y' -y, 
 
 P'R ^ ^' _ 2. 
 
 If we put 
 A = PP\ the distance of the points, we hav3, by Geometry, 
 A" = P'W + P'N'' + P'R 
 
 = {x' - xy + 0/' - yy + (.' - zy 
 
 = x'^ + y'' + ^" + ^^ + f-\- z' - Kxx' + yy'^r zz^). 
 Hence J = V(x''^^^xy~~-f~{y'~^y'-\- (z' - zy, (3) 
 
 To express the angles a, j3 and y which the line PP' 
 forms with the axes, we note that these angles are, by § 207, 
 equal to those which P'P forms with P'M, P'N and P' R 
 respectively. Thus we find 
 
 cos a 
 
 X' — X 
 
 cos ft = '^-j^-\ ^ 
 
 z' — z 
 COS y = —-^-. 
 
 (4) 
 
POSITION AND DIRECTION IN SPACE. 233 
 
 216. Problem. To express the angle hetiveen two lines 
 in terms of the angles which each of them forms with the co- 
 ordinate axes. 
 
 Let the two lines emanate from the origin, and put 
 
 a, (3, y, the angles which one line makes with tlie axes; 
 a' , yS', y' , the corresponding angles for the other line. 
 
 On each line take a point, namely, P on the one and P' 
 on the other, and put 
 
 r, r' , the distances of P and P' respectively from the origin. 
 
 The problem is solved by expressing the distance PP' \\\ 
 two ways: 
 
 I. The equations («) of § 214 give, for the co-ordinates 
 
 oi P'.^x — r cos a\ y — r cos /?; z — r cos y\ 
 of P'\ x' — t' cos a'\ y' — r' cos yS'; z' — r' cos /'. 
 
 Substituting these values in the expression of § 215 for 
 the distance of the points, 
 
 A- = 7*'^(cos*<a: + cos''/? + cos';/) 
 + r'XcosV' + cos'^/J' + cosV) 
 — 2rr'(cos a cos a' -\- cos y5 cos /?' + cos y cos y'). 
 
 The first two terms reduce to r^ + r" by {h). 
 
 II. If we put 
 
 V = the angle between the lines, 
 
 the lines r, r' and A will be the sides of a plane triangle of 
 which the angle opposite the side A is v. Hence, by Trigo- 
 nometry, 
 
 z/' = r' -f- ?'" — %rr' cos v. 
 
 III. Comparing the two values of A"^, we find 
 
 cos i; = cos <af cos a' + cos /? cos ^' + cos y cos y' , (5) 
 
 which is the required expression. 
 
 Cor. The condition that v shall be a right angle is 
 
 cos a cos a' + cos ^ cos yS' + cos y cos y' — 0, (G) 
 
 because, in oi-der that an angle shall be a right angle, it is 
 necessary and sufficient that its cosine shall be zero. 
 
234 GEOMETRY OF TUliEE DIMENSIONS. 
 
 317. Def. The direction-cosines of a line are the 
 cosines of the three angles which it forms with the co-ordi- 
 nate axes. 
 
 The direction-cosines are so called because they determine 
 the direction of the line. 
 
 Direction- Vectors. The three direction-cosines of a line 
 are not independent, because, when any two are given, the 
 third may be found by the equation 
 
 cos^a + cos^/? -|- cosV = 1- C^) 
 
 But the direction of the line may he defined ly any three 
 quantities proportional to its direct ion-cosi7ies. To show this, 
 let us put 
 
 I, m, n, any three quantities proportional to cos a, cos /?, 
 cos y respectively. 
 
 Because of this proportionality, we shall have 
 ? _ m _ w _ ^ 
 
 cos a cos fi ~ cos y ~ 
 whence 
 
 a cos a = /; o" cos /? = m; a cos y = n. 
 The sum of the squares of these equations gives, by (7), 
 (?» = r -f m= + n\ 
 I I 
 
 cos a = 
 
 cos /3 
 
 a 
 
 
 Vf 
 
 -f m^ 
 
 + 
 
 n"" 
 
 m 
 
 = 
 
 
 m 
 
 
 
 a 
 
 Vf 
 
 + w^ 
 
 + 
 
 n^ 
 
 n 
 
 
 
 
 n 
 
 
 
 (8) 
 
 cos r — - , 
 
 (? |//2 _|. ^yi" + n"" 
 
 Thus, when I, m and w are given, the angles a, /? and y 
 can be found, and thus the direction of the line is fixed. 
 
 The quantities I, m and n are called direction-vectors. 
 
 The direction of the line depends only upon the mutual 
 ratios of the direction-vectors, and not upon their absolute 
 values. For, if we multiply the three quantities Z, m and n 
 by any factor p, a will be multiplied by this same factor, 
 which will divide out from the equations (8), and thus leave 
 the values of a, /? and y unchanged. 
 
POSITION AND DIRECTION IN SPACE. 235 
 
 Cor, If the directions of two lines are given by the 
 direction-vectors 
 
 I, m and w, V, m' and n' , 
 respectively, the condition that they shall form a right 
 angle is 
 
 IV 4- mm' + nn' - 0. (9) 
 
 For, by substituting in tlie equation (6) the values of the 
 direction-cosines of the two lines given by (8), the condition 
 becomes 
 
 IV + mm' + nn' _ 
 
 6o' ~^' 
 
 from which (9) immediately follows. 
 
 218. Problem. To exiwess the square of the sine of the 
 angle hetiveen tivo lines in terms of their direction-cosines. 
 The result is derived from (5) by the form 
 sm^v = 1 — cos'^v = cos^a -\- cos'^/? + cos^;/ — cos^t'. 
 To simplify the writing we shall omit the letters cos, using 
 a, /3, y and «', y5', y'iov the direction-cosines of the respec- 
 tive lines. Then 
 sXn^v = a^ -\- fi' -\- y^ — Qo^'^v 
 
 = a'^ /5^-f y'- a' a"- /3'/3"- y'y'' 
 
 - 2aa'/3/3' - 2P/3'vy'- 2yy'aa' 
 = «»(1 _ ^-) 4. j3\l - ^-)+ y\l - y^) 
 
 -2(af«'/?y^' + etc.) 
 = a\r-^r yn + /5^(k"+ oc'^) + y\oc'^-\- D 
 
 -2{aa'/3/3' -^etc.) 
 = a'/^"-{- a''j3'- 2aa'p/3'-{- /5V'^+ /3'Y- ^^^'yy' 
 
 + y^a'^-^ y"a'- 2yy'aa' 
 = {ajS'- a'PY-{- {Pf- fiyy-\- {ya'- y'a)\ (10) 
 which is the required expression. 
 
 Cor. If tivo lines have the direction- cosines of the one 
 respectively equal to the corresponding ones of the other, the 
 lines are parallel. 
 For, if 
 
 a = a', ^ = /?', y = f, 
 
 the last equation reduces to 
 
 sin V = 0. 
 
236 GEOMETRY OF THREE DIMENSIONS. 
 
 EXERCISES. 
 
 1. If a line make equal angles with the co-ordinate axes, 
 what are these angles, and what angle does it form with the 
 co-ordinate planes? Aiis. 54° 44'. 1; 35° 15'. 9. 
 
 2. Find the direction-cosines of a line which makes equal 
 angles with the axes of X and Y, but double the common 
 value of those equal angles with the axis of Z, and show that 
 the angles may be either 90°, 90°, 180°; 45°, 45°, 90°; or 135°, 
 135°, 270°. 
 
 3. In a room 15 by 20 feet and 10 feet high, a line is 
 stretched from the northwest corner of the ceiling to the 
 southeast corner of the floor. Find its length, the angles 
 which it forms with the three bounding edges of the walls 
 and ceiling, and with the walls and ceiling. 
 
 4. What angles do lines having the following direction- 
 vectors form with the co-ordinate axes? — 
 
 Line A, 
 
 I = 1; 
 
 m = 2; 
 
 71 = 2; 
 
 Line B, 
 
 I = 3; 
 
 m = 2; 
 
 71 = 1; 
 
 Line 0, 
 
 I = 2; 
 
 m = 3; 
 
 71 = 4.; 
 
 Line D, 
 
 i^p; 
 
 m = 2p; 
 
 71 = Sjp. 
 
 5. If the direction-cosines of a line are proportional to the 
 fractions ^, -J, ^, what are the smallest integers which we can 
 employ as direction- vectors? 
 
 6. Find the values of the direction-cosines of a line which 
 satisfy the equation 
 
 cos a = 2 cos J3 = 3 cos y, 
 
 and the least integers which can be used as direction-vectors. 
 
 7. Find the direction-cosines of lines Joining the following 
 pairs of points: 
 
 (a) From tlie origin to the point (2, 3, 4); 
 
 (b) From the origin to the point (— 2, — 3, — 4); 
 
 (c) From the point (1, 1, 1) to the point (2, 3, — 1); 
 
 (d) From the point (1, 2,- 3) to the point (-1-3, 3). 
 
 If the order of the points of each pair be reversed, what effect 
 will this change have on the direction-cosines? 
 
POSITION AND DIRECTION IN SPACE. 
 
 237 
 
 8. What angle is formed by the two lines passing from 
 the origin to the points (1, 1, 2) and (2, 3, 4) respectively? 
 
 9. Find the angle whose vertex is at the point (2, 3, 4), 
 and whose sides pass through the points (1, 2, 3) and (3, 5, 5). 
 
 10. What angle is contained by two lines whose direction- 
 vectors are : 
 
 Line A, / = -|-l; m = -\- 3; n = — 5; 
 Line B, I = - 3; m = + 2; n = -\- 1. 
 
 219. Transformation of Co-ordinates. 
 
 Case I. Transformation to a new system whose axes are 
 parallel to those of the first system. Let the co-ordinates of a 
 point P referred to the old system be x, y and z, and let the 
 co-ordinates of the new origin referred to the old system be 
 a, ^ and c. It is now required to express the co-ordinates 
 x', y', z' of P referred to the new system. 
 
 Because the new and old co-ordinate planes are parallel, 
 the perpendiculars dropped from the point P upon corre- 
 sponding planes will be coincident, and that portion of a 
 perpendicular intercepted between the parallel planes will be 
 a, b or c according as the plane is YZ, ZX or XY. The 
 difference between the co-ordinates will therefore be equal to 
 these same quantities, and we shall have 
 
 X' :=^x -a\ y =y -^, 
 or x =x' ^a-, y = / + &; 
 
 Z 
 220. Case II. Transforma- 
 tion to a neiu rectangular system 
 having the same origin hut differ- 
 ent directions. Let 0-XYZ be 
 the axes of the old system, OX' 
 any axis of the new system, and 
 P a point whose co-ordinates 
 are to be expressed in both sys- 
 tems. ( 
 From P drop PQA. plane XY, 
 From Q drop QRL axis OX. 
 
 (11) 
 
 axis OZ. 
 
238 QEOMETBY OF THREE DIMENSIONS. 
 
 Then, calling x, y and z the co-ordinates of P referred to 
 the old system, we shall have, by § 213, 
 
 X = OR', 
 
 y = RQ\ 
 
 z = QP. 
 
 From the points P, Q and R drop perpendiculars upon the 
 new axis OX', meeting it in the points P', Q' and R' . Let 
 us then put 
 
 or, /?, /, the angles which OX' , the axis of the new sys- 
 tem, makes with the respective axes of X, Y and Z in the 
 old system. We shall then have 
 
 OR' = OR cos XOX' = X cos a\ 
 
 also, because RQ \\ OY, 
 
 R'Q' = RQ cos YOX' = y cos /3; 
 also, because QP \\ OZ, 
 
 Q'P' = QP cos ZOX' = z cos y. 
 
 Now, the line OP' is the algebraical sum of the three seg- 
 ments OR', + R'Q'y + Q'P', each segment being taken posi- 
 tively or negatively according as the angle a, /3 ov y is acute 
 or obtuse. 
 
 Hence 
 
 OP' = X cos a -\- y COB p -\- z cos y. (12) 
 
 If we suppose OX' to represent the new axis of X, then 
 OP' will be the co-ordinate x referred to the new axis, which 
 we call x'. In the same way we have y' and z' when OX* 
 represents the corresponding axes. If, therefore, we put 
 
 (X',X),(X', Y),{X',Z) the angles made byX with X, F& Z; 
 (F',X),(V',F), (T',^) the angles made by F' wij:h X, F& Z; 
 \Z',X), (Z', Y), (Z',Z) the angles made by Z' with X, Y& Z, 
 
 we shall have 
 
 x' = x cos (X',X) + ^cos (X', Y) + z cos (X',Z); ) 
 ?/'=a;cos(r',X) + ?/cos(F',r)4-^ cos (F',Z); [ («) 
 z' = x cos (^',X) -h 2/ cos (Z', Y)-\-z cos (Z', Z). ) 
 
x' 
 
 = 
 
 X COS a 
 
 y' 
 
 =: 
 
 X cos 
 
 a' 
 
 z' 
 
 = 
 
 X COS 
 
 a" 
 
 POSITION AND DIRECTION IN SPACE. 239 
 
 The reliition of the symbols {X',X)y etc., to the symbols 
 X, y, z and a:', ?/', 2;', which is readily seen, renders these equa- 
 tions easy to write. But the subsequent management of the 
 equations will be more simple if we retain the symbols a, j3 
 and y, putting 
 
 a, ^ and y for {X',X),{X' ,Y) and (X',Z); 
 a', f3' and/ for (r',.r),(F', F) and (F',Z); 
 a", ft" and y" for {Z',X),{Z\ Y) and (Z', Z). 
 
 The equations (a) will then be written 
 
 -f y COS y5 -{-5; cos ;/; j 
 
 -i-y cos /3' -\-z cos ;r'; [■ (13) 
 
 + 2^ cos /5" 4- 2^ cos y''. ) 
 
 Each set of these cosines must separately satisfy the equa- 
 tion (7), which gives the first three equations written below. 
 The last three are obtained by the consideration that, by § 216, 
 the cosine of the angle between the axes of X' and Y' is 
 
 cos a cos a' -\- cos ft cos ft' -j- cos y cos y'. 
 
 But, because the new axes are rectangular, this cosine must 
 be zero, as must also be the cosines of the angles between Y' 
 and Z', and between Z' and X\ Thus we have the six equa- 
 tions of condition, 
 
 cos^<^ + cos^/? + cosV = 1; 
 
 cos^a' + cos^/5' + cos''/' = 1; 
 
 cosV + co^'ft" + cosV" = 1; 
 
 cos a cos a' -j- cos ft cos ft' -\- cos y cos y' = 0; 
 
 cos «' cos a" + cos /5' cos /5"-|- cos ;k' cos ;/"= 0; 
 
 cos a" cos a -|- cos /5" cos ft + cos /" cos y = 0. 
 
 There being six separate equations of condition between 
 the nine cosines, it follows that all nine of them can be ex- 
 pressed in terms of some three independent quantities. How 
 this can be done we shall show hereafter. 
 
 321. We next remark that we can express the co-ordi- 
 nates X, y and z in terms of x', y' and z' , by reasoning exactly 
 as we have reasoned in the reverse case, thus obtaining 
 
 (14) 
 
g40 GEOMETRY OF THREE DIMENSIONS. 
 
 X = x' cos cc -\- ff cos «' + z' cos a"; \ 
 
 ?/ = .t'cos/? + ?/'cos/?'+2'cos/5"; V (15) 
 
 z = x' cosy -\- if cos y' + 2;' cos y". ) 
 
 We can also derive the first of these equations directly 
 from (12) by multij)lying the first by cos a, the second by 
 cos a' and the third by cos a", and adding, noting the ap- 
 plication of the results of § 216 to the angles formed by the 
 axes. 
 
 Continuing the reasoning, we are led to the six equations 
 of condition, 
 
 cos^a: + cosV + cosV = 1; 
 
 cos'/5 + cos^/J' + cos^/J" = 1; 
 
 cos";/ + cosY' + cos'^;/" = 1; 
 
 cos a cos /? 4- cos a' cos /?' + cos a" cos /3" = 0; 
 
 cos /3 cosy ^ cos /5' cos y' + cos /5" cos y" = 0; 
 
 cos y cos a -{- cos 7' cos a' + cos ;k" cos a" = 0. 
 
 (16) 
 
 In reality these equations are equivalent to the equations 
 (14), and the one set can be deduced from the other by alge- 
 braic reasoning, without any reference to co-ordinates. 
 
 222. Polar Co-ordinates in Space. In space, as in a 
 plane, the position of a point is determined when its dirediori 
 and distance from the origin are given. 
 
 In space the direction requires two data to determine it. 
 These data may be expressed in various ways, of which the fol- 
 lowing is the most common. We take, for positions of refer- 
 ence: 
 
 1. A fixed plane, called the /?m^amew^aZjt??awe. For this 
 the plane of XY in rectangular co-ordinates is generally 
 chosen. 
 
 2. An origin or pole, 0, in this plane. 
 
 3. A line of reference, for which we commonly choose the 
 axis of X. 
 
 Let P be the point whose position is to be defined. We 
 first have to define the direction of the line OP. From any 
 point P of this line drop a perpendicular PQ upon the fun- 
 
POSITION AND DIRECTION IN SPACE. 
 
 241 
 
 damcntal plane, and join 0(2. The direction of OP is then 
 
 defined by the following two r^ 
 
 angles: 
 
 (1) The angle P0(> which 
 OP forms with its projection 
 OQ] that is, the angle be- 
 tween OP and the plane. 
 
 (2) The angle X0§ which 
 the projection of OP makes 
 with OX. 
 
 It will be remarked that 
 the planes of these two angles are perpendicular to each 
 other. 
 
 To show that these two angles completely fix the direction 
 of OP, we first remark that when the angle XOQ is given, 
 the line OQ \s fixed. 
 
 Next, because PQ is perpendicular to the plane, the point 
 P and therefore the line OP must lie in the plane ZOQ, 
 which is fixed because its two lines O^and OQ are fixed. If 
 the angle QOP in this (vertical) plane is given, there is only 
 one line, OP, which can form this angle. 
 
 Hence the direction of the line OP is completely deter- 
 mined by the two angles XOQ and QOP', and when the dis- 
 tance OP is given, the point P is completely fixed. 
 
 We use the notation 
 
 cp, the angle QOP, or the elevation of OP above the 
 plane XOY. We may call this angle the latitude of P. 
 
 A, the angle XOQ which OQ, the projection of OP, 
 makes with OX. We may call this angle the longitude of P, 
 
 r, the length of OP. 
 
 Because the quantities cp, \ and r completely fix the posi- 
 tion of P, they are called the polar co-ordinates of P in 
 space. 
 
 233. Relation of the Preceding System to Latitude and 
 Longitude. For another conception of the angles q) and A, 
 pass a sphere around as a centre, and mark on its surface 
 the points and lines in which the lines and planes belonging 
 to the preceding figure intersect it. Then 
 
242 
 
 GEOMETRY OF THREE DIMENSIONS. 
 
 The fundamental plane OXQ intersects the spherical sur- 
 face in the great circle XQY\ 
 
 The line OX intersects it in X; 
 
 The line OQ intersects it in Q-, 
 
 The lines OP and OZ intersect it in P and Z. 
 
 We therefore have 
 
 Angle XOQ measured by arc XQ\ 
 Angle QOP measured by arc QP, 
 
 If now we imagine this sphere to be the earth, the great 
 circle XP" to be its equator, Z to be one of the poles, and P 
 any point on its surface, then 
 
 The arc QP or the angle QOP is the latitude of P; 
 
 The arc XQ or angle XOQ is the longitude of P, counted 
 from ZX as a prime meridian. Thus the angles we have 
 been defining may be described under the familiar forms of 
 longitude and latitude. 
 
 224, Peoblem. To transform the position of a point 
 from rectangular to polar co-ordinates, and vice versa. 
 
 Comparing the definitions of rectangular and polar co- 
 ordinates, we put, for the point P in § 222, 
 
 PQz=z=: OP sin ^; 
 OQ = OP cos (p. 
 
POSITION AND DIRECTION IN SPACE. 243 
 
 Now, supposing a perpendicular dropped from Q upon OX, 
 this perpendicular will be the ordinate y, and will meet OX 
 at the distance x from the origin. Thus, 
 
 x= OQ cos XOQ = OQ cos ?i; 
 y=OQ sin XOQ = 0^ sin X. 
 
 Putting OP = r, and substituting for 0§ its value, we have 
 
 ic = r cos cp cos A; j 
 
 y = r cos 99 sin A; >• (17) 
 
 z = r sm cp'j ) 
 
 which are the required equations. 
 
 Cor. The direction-cosines of the line OP in terms of cp 
 and A are 
 
 cos a = cos q) cos A; j 
 
 cos /? = cos cp sin A; >• (18) 
 
 cos y = sin cp. ) 
 
 235, The result stated in § 220, that the nine direction- 
 cosines of one system of rectangular axes with respect to 
 another system can be expressed in terms of three independ- 
 ent quantities, may now be proved as follows: 
 
 1. Let OP be any one accented axis, say JT'; the direction- 
 cosines of this axis are expressed in terms of two angles, 
 (p and A, by (18). 
 
 2. Imagine a plane =31, passing through 0, §223, per- 
 pendicular to OP. This plane M will be completely deter- 
 mined by the direction OP; whence the line = iV'in which it 
 cuts the fundamental plane XY will also be determined. 
 
 3. The new axis V may lie in any direction from the 
 point in the plane M. One more angle = ?/? is required to 
 determine this direction, and for this angle we may take the 
 angle which F' forms with the line iV. 
 
 4. The direction of the axis Z' is then completely fixed, 
 because it must lie in the plane 31 and make an angle of 90° 
 with Y\ 
 
 Thus, cp, A and tp completely determine the directions of 
 the three new axes. 
 
244 GEOMETRY OF TUREE DIMENSIONS. 
 
 EXERCISES. 
 
 1. If, in the figure of § 223, the co-ordinates of the point 
 
 P are 
 
 a; = 27, y = l^, ^ = 17, 
 
 find its polar co-ordinates r, q) and \. 
 
 Method of Solution. The quotient of the first two equations (17) gives 
 
 tan A = ^, 
 
 X 
 
 from which A. is found. Then we find sin X or cos X or both, and com- 
 pute 
 
 X y 
 
 r cos cp = r- = - — Y' 
 
 cos X sin X 
 
 Next we have 
 
 z 
 
 tan cp = , 
 
 r cos q) 
 
 from which we find <p. Then 
 
 z _ r cos (p 
 "" siu ^ " cos cp ' 
 
 2. Supposing the radius of the earth to be 6369 kilometres, 
 the longitude of New York to be 74° west of Greenwich, and 
 its latitude to be 40° 32', it is required to find the rectangular 
 co-ordinates of New York referred to the following system of 
 axes having the earth's centre as the origin: 
 
 Xin the equator, and on the meridian of Greenwich. 
 Y in the equator, in longitude 90° east of Greenwich . 
 Z passing through the North Pole. 
 
 3. If, in the figure of § 222, we take a point P' whose 
 latitude is the same as that of P and whose longitude is 90° 
 greater than that of P, it is required to express the angle 
 POP\ 
 
 4. If the angle <p is negative, within what region will the 
 point P be situated? 
 
 5. If we take a point P' whose latitude is cp and whose 
 longitude is A + 180°, how will it be situated relatively to P, 
 and what will be the angle POP'? 
 
 6. If we take a point P' for which 
 
 (p' = 180° - cp, 
 
 r =x-^ 180°, 
 
 show that this point will be identical with P. 
 
CHAPTER II 
 
 THE PLANE. 
 
 226. Introductory Considerations on the Loci of Equa- 
 tions. If the values of the three co-ordinates of a point are 
 not subject to any restriction, the point may occupy any posi- 
 tion in space. Kestrictions upon the position are algebraically 
 expressed by equations of condition between the co-ordinates. 
 Let us inquire Avhat will be the locus of the point when the 
 co-ordinates are required to satisfy a single equation of condi- 
 tion. By means of such an equation we may express any one 
 of the co-ordinates, z for example, in terms of the other two, 
 the form being 
 
 We can now assign any values we please to x and y, and 
 for each pair of such values find the corresponding value of z. 
 To each pair of values of x and y will correspond a certain 
 point on the plane of JTY. If at this point we erect a per- 
 pendicular equal to z, the end of each perpendicular will be a 
 point whose co-ordinates satisfy the equation. 
 
 We may conceive these perpendiculars to become indefi- 
 nitely numerous and indefinitely near each other, thus tending 
 to form a solid. Their ends will then tend to form the sur- 
 face of this solid. But these ends are the locus of the equation 
 (a). Hence 
 
 The locus of a single equation of condition among the co- 
 ordinates is a surface. 
 
 If a second equation is required to subsist among the co- 
 ordinates, the locus of this equation will be a second surface. 
 
 If the co-ordinates are required to fulfil both conditions 
 si7nultaneously J then the point must lie in both surfaces; that 
 
246 GEOMETRY OF THREE DIMENSIONS. 
 
 is, it must lie on the line in which the surfaces intersect. 
 Hence 
 
 The locus of two simultaneous equations betioeen the co- 
 ordinates is a line, 
 
 2^7. To find the Eq^iation of a Plane. The property of 
 a plane from which the locus can be most elegantly deduced is 
 this: If on any line which intersects the plane perpendicularly 
 we take two points, A and B, equidistant from the point of 
 intersection and on opposite sides, then every point of the 
 plane will be equidistant from A and By and no point not on 
 the plane will be equidistant. 
 
 Let us then drop from the origin a perpendicular upon the 
 plane, and continue it to a distance on the other side equal to 
 its length, and let P be the point at which it terminates. 
 The condition that a point shall lie on the plane will then be 
 that it shall be equidistant from the origin and from P, 
 
 Let us put 
 
 a, h, c, the co-ordinates of P ; 
 
 X, y, z, the co-ordinates of any point on the plane. 
 
 Then, by §§ 214, 215, the squares of the distances of (x, y, z) 
 from the origin and from P will be respectively 
 
 x^^y'-^rz^ 
 and {x - ay -\- {y - hy + (^ - cy. 
 
 Developing and equating these two expressions, we find, for 
 the required equation, 
 
 2ax + Uy + 2cz = a' + b' -\- c\ 
 
 To reduce this equation, let us put 
 
 Py the length of the perpendicular dropped from the origin 
 upon the plane; that is, one half the line from the origin to P ; 
 a, /3, y, the angles which this perpendicular makes with 
 the respective axes of JT, Y and Z. 
 
 We shall then have OP = 2p, and the values of a, I? and 
 c will be, by § 214, 
 
 a = 2p cos a; 
 b = 2p cos /?; 
 c = 2p cos y. 
 
THE PLANE. 247 
 
 Substituting these values in the equation of the plane, reduc- 
 ing, and remarking that 
 
 cos''^ + cos'/? + cos';/ = 1, 
 the equation of the plane becomes 
 
 X cos a -{- y co^ ^ -\- z 0,0^ y — p — 0, ( 1 ) 
 
 which is called the normal equation of the plane. 
 
 228. The A^igles a, fi and y. As we have defined the 
 angles «, /? and y, they are the angles which the perpen- 
 dicular p forms with the co-ordinate axes. It is shown in 
 Solid Geometry that the angle between any two planes is equal 
 to that between any two lines perpendicular to them. Because 
 
 Plane YZ ± axis X, 
 
 Plane (1) ± line p, 
 
 . • . plane (1) makes the angle a with the plane YZ. Hence 
 
 we may define a, ft and y as the angles which the plane mahes 
 
 with the co-ordinate planes YZ, ZX and XY respectively. 
 
 One restriction upon this proposition is necessary. Two 
 supplementary angles are formed by any two planes, so that, 
 in the absence of any convention, we should say that the 
 angle between the planes is either equal or supplementary to 
 that between the lines. The best way of avoiding ambiguity 
 is to choose, for the angle between the planes, the angle be- 
 tween the perpendiculars dropped from the origin upon the 
 planes. 
 
 If the angles a, f3 and y are the same for several planes, 
 these planes, being perpendicular to the same line, are paral- 
 lel. Hence they may, in a certain sense, be said to have the 
 same direction. We may therefore call cos a, cos y^ and cos y 
 the direction-cosines of the plane. They are also the direc- 
 tion-cosines of the perpendicular dropped from the origin 
 upon the plane. 
 
 229. Theorem I. Every equation of the first degree be- 
 iween rectangular co-ordinates in space is the equation of 
 some plane. 
 
 Proof Let the equation be 
 
 Lx^My^Nz-\-D = 0. 
 
248 GEOMETRY OF THREE DIMENSIONS. 
 
 Divide this equation by VU -{- M'^ -\- iV% and determine 
 three angles, a, /? and y, by the equations 
 
 L 
 
 cos a 
 
 COS p 
 
 (2) 
 
 COS y — — — . 
 
 This will always be possible, because each of these cosines 
 is less than unity. Because these cosines fulfil the condition 
 
 (7), B 217, we can draw a line of length = , 
 
 from the origin making the angles a, § and y with the sev- 
 eral co-ordinate axes. Through the end of this line pass a 
 plane perpendicular to it. Then, by the last section, the equa- 
 tion of this plane will be 
 
 a; cos a -I- V cos /? + 2? cos 1/ -] 3= — 0, 
 
 an equation which becomes identical with that assumed in 
 the hypothesis by clearing of denominators and substituting 
 from (2). We may therefore put the theorem in the following 
 more specific form: 
 
 Every equation of the form 
 
 Lx^ My -^ Nz-\- D = 
 
 represents a certain defitiite plane, namely y the plane passing 
 perpendicularly through the end of that line tuhich 
 
 emanates from the origin; 
 
 makes with the axes angles whose cosines are 
 
 L M ^ N 
 
 — and 
 
 respectively; 
 
 and has the lenqth —t=~ — -^ — 
 
 VU + M' + N' 
 
THE PLANE. 249 
 
 230. Notation, By " the plane (L, M, N, D)" we mean 
 '' the plane whose equation is 
 
 Lx + Ml/ -{- JVz-{-I) = 0." 
 
 Vef. An equation of a plane in which the four quantities 
 Z, if, a and D are all independent is called the general 
 equation of a plane. 
 
 The general equation may be considered as related in two 
 ways to the normal or other special forms of the equation. 
 
 I. The special forms are cases in which certain relations 
 exist among the quantities L, M, iVandi). For example, the 
 normal form is the special case in which U -\- M^ -{-N"^ = 1. 
 
 Whenever we find this condition satisfied, we know that 
 the equation is in the normal form. 
 
 II. The general equation may always be reduced to the 
 normal form by dividing by V U -f if ^ -J- N"^. 
 
 Direction- Vectors. From the equation (2) it is seen that 
 L, M and N may be taken as the direction-vectors of any line 
 perpendicular to the plane, because they are severally equal to 
 the direction-cosines of such a line multiplied by the common 
 factor \/{U -+- M"^ + N^)- Hence we conclude: 
 
 The equation of every plane perpendicular to a line whose 
 direction-vectors are I, m and n may he luritten in the form 
 
 lx -f my -\- nz -\- d = 0', 
 
 and, conversely, /or the direction-vectors of any line perpen- 
 dicular to the plane {L, M, N, D) may he taTcen L, M and N. 
 
 231. Special Positions of a Pla7ie. If one of the co- 
 efficients L, M, or ^Y vanishes, the cosine of the angle which 
 the plane makes with the corresponding co-ordinate plane will 
 also vanish; that is, the plane will be i^erpendicular to the co- 
 ordinate plane, and therefore parallel to the axis of that plane. 
 Hence a7i equation of the first deyree hetioeen two only of the 
 co-ordinates represents a 2)lane parallel to the axis of the miss- 
 ing co-ordinate. 
 
 For example, the locus of 
 
 Lx-\-My^D = 
 is a plane parallel to the axis of Z. 
 
250 GEOMETRY OF TUREE DIMENSIONS. 
 
 It follows that if two co-ordinates are missing, the locus 
 will be parallel to the common plane of the missing co-ordi- 
 nates. For example, the locus of 
 
 Lx -^ D = 
 D 
 
 will be a plane perpendicular to the axis of X and parallel to 
 the plane YZ. 
 
 233. Lines and Points coujieded with a Plane. The 
 following lines and points are determined by every plane: 
 
 I. The three lines in which it intersects the co-ordinate 
 planes. 
 
 II. The three points in which it intersects the co-ordinate 
 axes. 
 
 III. The foot of the perpendicular from the origin upon 
 the plane. 
 
 When we include among possible lines and points the lines 
 and points at infinity, the above three lines and four points 
 will always be determinate. 
 
 Def. The lines in which a plane intersects the co-ordinate 
 planes are called traces of the plane. 
 
 The distances from the origin to the three points in which 
 a plane cuts the co-ordinate axes are called the intercepts 
 of the axes by the plane. 
 
 233. Problem. To find the equations of the traces of a 
 plane. 
 
 The trace of the plane upon the plane of YZ is simply 
 those points of the plane for which ic = 0. Hence, if we put 
 X = in the equation of a plane, we have the equation of its 
 trace upon YZ. Therefore, in the general equation 
 
 Lx-^ My -\- Nz-{- D= 0, 
 
 the equations of the traces upon the co-ordinate planes are: 
 
 On YZ, My + Xz ^ D = 0-, 
 On ZX, Lx -\- Nz -\- D = (); 
 On XY, Lx + My+D^ 0. 
 
THE PLANE. 
 
 251 
 
 These equations, representing lines upon planes, can be dis- 
 cussed like the equations of lines in Plane Analytic Geometry. 
 
 234. Problem. To express the lengths of the intercepts 
 of the axes hy a plane. 
 
 At the point where the plane cuts the axis of Xwe have 
 y = and 2; = 0. Hence the intercept is the value of x cor- 
 responding to zero values of y and z, and so with the other 
 co-ordinates. Thus: 
 
 Intercept on X = 
 Intercept on Y = 
 Intercept on Z = 
 
 n 
 
 D 
 
 D 
 
 N' 
 
 (4) 
 
 Scholium. Each of the traces necessarily meets the other 
 two on the several co-ordinate axes, and their points of meet- 
 ing are those in which the plane cuts the axes. Hence the 
 traces form a plane triangle of which the points in which the 
 plane intercepts the axes are the vertices. 
 
 Each of the sides of this triangle is the hypothenuse of a 
 right triangle of which the sides containing the right angle 
 are the intercepts upon the axes. 
 
 The relations between the sides and angles of these tri- 
 angles, considered individually, may be investigated by the 
 methods of Plane Trigonometry. 
 
 235. Problem. To express the equation of a plane 171 
 terms of its intercepts upon the axes 
 
 Let us put 
 
 a, I, Cy the intercepts on the axes of X, Y and Z respec- 
 tively. 
 
 Then 
 
 a = — 
 
 D 
 
 ■ -'-',■. 
 
 D 
 
 .. .,.--.. 
 
 D 
 
 c 
 
 (5) 
 
252 GEOMETRY OF THREE DIMENSIONS. 
 
 Substituting these values of L, M and N in the general equa- 
 tion, it reduces to 
 
 1+1+^ = ^' («) 
 
 which is the required equation. 
 
 EXERCISES. 
 
 1. Write the equation of that plane for which the co- 
 ordinates of the foot of the perpendicular from the origin 
 upon the plane are 1, 2, and 3. 
 
 2. If, in the general equation of a plane, the coefficients 
 L, Jtf and iV^are all equal, what angle will the perpendicular 
 make with the co-ordinate axes, and Avhat angle will the 
 plane make with the co-ordinate planes? 
 
 3. The equation of a plane being 
 
 3a; + 4?/ - 122; = 26, 
 
 it is required to reduce it to the normal form to find the 
 angles which it forms with the co-ordinate axes, the equations 
 of its traces upon the co-ordinate planes, the lengths of its in- 
 tercepts upon the co-ordinate axes, the lengths of its traces 
 between these intercepts, and its least distance from the 
 origin. 
 
 4. The intercepts being in the proportion 1:2:3, what 
 are the cosines of the angles which the perpendicular upon 
 the plane makes with the axes? 
 
 5. Show that the inverse square of the perpendicular from 
 the origin upon a plane is equal to the sum of the inverse 
 squares of its intercepts; i.e., 
 
 1 = 1 + 1 + 1. 
 
 p a c 
 
 6. Show the corresponding relation between the two sides 
 of a right triangle and the perpendicular from the vertex 
 upon the hypothenuse. 
 
 7. Express the lengths of the sides of the triangle formed 
 by the traces of a plane in terms of the intercepts, and prove 
 that the sum of the squares of the sides is twice the sum of 
 the squares of the intercepts. 
 
THE PLANE. 253 
 
 8. A plane cuts traces whose lengths between the axes are: 
 On plane YZ, u\ on plane ZX, v; on plane XY, w. 
 
 Find the lengths of the intercepts and the equation of the 
 plane in terms of s^ = ^(u^ + ^^ + w^). 
 
 Ans. — ^= + -JL^ + -=^= = 1. 
 
 9. Find the angle included between the planes 
 
 and x-y-{-2z = b. (Comp. §§ 21G, 230) 
 
 236. Plane satisfying Given Conditions. If a plane is 
 required to satisfy a condition, that condition can be expressed 
 as an equation between the constants L, M, N, D, which de- 
 termine the position of the plane. By means of this equation 
 one of the constants can be eliminated from the equation of 
 the plane, and the condition will then be fulfilled for all values 
 of the remaining constants. 
 
 If two conditions are given, two constants can be elimi- 
 nated; if three, all the constants. For, although the general 
 equation of the plane contains four constants, it depends only 
 on the three ratios of any three of these constants to the 
 fourth. In fact, we can always reduce the general equation to 
 the form (6) 
 
 a^h^ c ^' 
 which contains but three arbitrary constants. 
 
 237. Problem. To find the equation of a plane passing 
 through a given point. 
 
 Let {x\ y', z') be the given point. In order that the 
 plane {L, M, JV, D) may pass through this point, its constants 
 must satisfy the condition 
 
 Lx' -f My' 4- JV>/ + i) = 0, {a) 
 
 which gives 
 
 D=- (L:c' + J// + iW). 
 
 Substituting this value of D in the general equation, the 
 latter becomes 
 
 Lx + My -\-Nz- {Lx' + My' + Nz') = 0, (7) 
 
254 QEOMETEY OF THREE DIMENSIONS. 
 
 or, in another form, 
 
 X(:, - X') + Miy - y') + N{z - z') = 0. (8) 
 
 Remark 1. In these equations we may assign any values 
 we please to L, M imd iV, witliout the plane ceasing to pass 
 through the point {x\ y', z'), as is evident from (8). 
 
 Remark 2. If we had two equations of the form (a), we 
 could eliminate two of the constants, say N and D, and L 
 and if would still remain. If we had three equations, we 
 could eliminate three constants, M, iV^and D for example. 
 That is, we could, by solving the equations, express M, iV^and D 
 in terms of L. Substituting these values in the general equa- 
 tion, the latter would, it would seem, still contain the constant 
 L. But, in reality, L would enter only as a factor of the 
 whole equation, and would therefore divide out. Hence, when 
 we eliminate any three of the four constants, the fourth drops 
 out of itself without the introduction of any further condition. 
 
 Relations of Two or More Planes. 
 
 238. Parallel and Perpendicular Planes. 
 
 Theorem II. If, in the equations of any tiuo planes 
 Lx -f My -\- Nz -\- D ^ 0, 
 L'x + M'y + N'z + i)' z= 0, 
 tlie direction-vectors L', M' and N' are proportional to L, M 
 and N respectively, the two planes are parallel. 
 
 Proof. The direction-vectors of the perpendiculars from 
 the origin upon the planes being proportional, the direction- 
 cosines are equal (§ 217), and these perpendiculars are co- 
 incident (§ 218). Hence the planes are perpendicular to the 
 same line and therefore parallel. Q. E. D. 
 
 Problem. To find the condition that two plajies given ly 
 their equations shall he perpendicular to each other. 
 
 Let the planes be (Z, M, N, D) and {U, M', N', D'). 
 
 The planes will be perpendicular when the perpendiculai s 
 from the origin are perpendicular to each other. The con- 
 dition is, from eq. (9) of § 217, 
 
 LL' -f MM' + NN' = 0. 
 
THE PLANE. 255 
 
 239. Notation. I. We use tlie symbols P, P', P", etc., 
 Q, Q' , etc. etc., to signify functions of the co-ordinates of 
 the first degree. For example, 
 
 P = Lx -\- My + Nz + D', 
 P' = L'x + M'y + N'z + i)'; 
 etc. etc. etc. 
 
 II. By the expression "the plane P" we mean the plane 
 whose equation is P = 0. 
 
 240. Theorem III. If in a function P we substitute 
 for X, y and z the co-ordinates x' , y' and z* of a point, P will 
 then express the distance of that point from the plane P = 
 multiplied ly the factor V L" + if ^ -}- N^. 
 
 Proof. Let us pass through the point (re', y', z') a plane 
 A parallel to the plane P. The equation of this plane will 
 be (§ 237) 
 
 Lx + My -\-Nz- {Lx' + My' + Nz') = 0. 
 
 The term independent of x, y and z is —{Lx'-^My'-^Nz')y 
 which takes the place of D in the general equation. Hence 
 the perpendicular distance of this plane A from the origin is 
 
 Lx' 4- My' + Nz' 
 
 P 
 
 VU -\-M'' '-{-N^ 
 
 The perpendicular distance of the plane P from the origin is 
 (§229) 
 
 --D 
 
 Because the point {x', y', z') is in the plane A \\ P, the 
 distance of (x', y', z') from the plane P is equal to the con- 
 stant distance between the planes, and hence to the difference 
 p — p' between the perpendiculars. Hence 
 
 Distance of (:.', y', z') from P = L^±BL±M±^-, 
 or 
 
 Lx' + My' -f- Nz' + P = Distance x ^ U + if ^ ^N\ 
 
 Q. E. D. 
 
256 GEOMETRY OF THREE DIMENSIONS. 
 
 Cor. If the equation is in the normal form, we shall have 
 
 and tlie expression P will then represent the distance of the 
 point whose co-ordinates appear in it from the plane P = 0, 
 
 EXERCISE. 
 
 Show that the angle e between the two planes 
 
 Lx + My-]- N'z-^D = and L'x + M'y -\- N'z-rD' = 
 
 is given by the equation 
 
 LL' + MM' + iVN' 
 
 cos s = - - ' 
 
 i/L' -i-M'-\- iV'-' V U' -i- M'' -{- N'' 
 
 241. Theorem IV. If P = and P' = ie the equa- 
 tions of any two planes, and A and X' constants, the equation 
 
 XP 4- rp' = (a) 
 
 will he the equation of a third plane intersecting the other tivo 
 in the same line. 
 
 Proof. I. The expression XP + X'P' is of the first degree 
 in X, y and z. Therefore the equation is that of some plane. 
 
 II. Every set of values of the co-ordinates x, y and z 
 which simultaneously satisfy both equations P = and P' = 
 also satisfy equation {a). The co-ordinates which satisfy both 
 equations are those of their line of intersection (§ 226). There- 
 fore these co-ordinates also satisfy («); whence the line lies in 
 the plane XP -\- X'P', which proves the theorem. 
 
 Cor. If three functions, P, P' and P", are such that it 
 is possible to find three constant coefiicients, A, A' and A", 
 which lead to the identity 
 
 AP + A'P' + A"P"eO, 
 the three planes P, P' and P" intersect in the same line. 
 
 Theorem V. If Q = and Q' = are the eqiiations of 
 two planes in the normal form, the equal io7is 
 
 Q -f (2' = and Q - Q' = 
 
THE PLANE. 257 
 
 represent the planes which Used the dihedral angles formed hy 
 the plane Q and Q' . 
 
 Proof, Because the expressions Q and Q' represent the 
 distances of the point (re, ?/, z) from the planes Q and §' 
 (§ 240), it follows that the equation 
 
 q^Q' or § - §' = 
 
 will express the condition that the said point is equally dis- 
 tant from the planes Q and Q' . Hence it lies upon the plane 
 bisecting the angle formed by Q and Q' , and this plane is the 
 locus of the equation Q — Q' = 0. 
 
 The equation Q -\- Q' = is equivalent to Q = — Q' , 
 and asserts that the point (x, y, z), if on the positive side of 
 the one plane, is on the negative side of the other at an equal 
 distance. Therefore it bisects the adjacent dihedral angle. 
 
 Cor. In the case supposed, the two planes Q — Q^ and 
 (3 -f C' ^^6 perpendicular to each other because they are the 
 bisectors of adjacent angles. 
 
 Theorem VI. If A, A' a7id A" are constant coefficients, 
 the equation 
 
 AP + A'P'+ A"P" = {h) 
 
 represents a plane passing through the common point of inter- 
 section of the planes P, P' and P" , 
 
 Proof. In the same way as with Theorem IV, it is 
 shown (1) that the equation is that of a plane, and (2) that 
 the co-ordinates of any and every point common to the three 
 planes P, P' and P" satisfy equation {Jj). Now, because any 
 three planes have one point common (which may be at infin- 
 ity), the point common to P, P' and P" lies on the plane 
 {h). Q. E. D. 
 
 Cor. If four functions, P, P', P" and P'", are such that 
 an identity of the form 
 
 \P _|_ VP' -f V'P" + V"P"' = 
 
 is possible, the four planes P, P', P" and P'" will intersect 
 in a point. 
 
258 GEOMETRY OF THREE DIMENSIONS. 
 
 242. Bisectors of Dihedral Angles. The foregoing 
 principles enable us to prove many elegant relations among 
 the planes which bisect the dihedral angles of a solid, or of a 
 solid angle. Let 
 
 Q = 0, Q' = 0, e" = 0, 
 
 be the equations of any three planes in the normal form. 
 Since any three planes meet in a point, they may be consid- 
 ered as forming a dihedral angle at that point. 
 
 The bisecting planes of the three dihedral angles formed 
 by the planes Q, Q' and §" will be (§ 241) 
 
 Q - Q' = = P; 
 Q' _ §" ^ = P'; 
 Q'' - Q = E P". 
 
 These functions satisfy the condition 
 
 P -^ P' -\- P'' = 0. 
 
 Therefore the three hisecting planes of a dihedral angle in- 
 tersect on a line. 
 
 Placing the centre of a sphere at the vertex of the dihe- 
 dral angle, and considering the spherical triangle formed by 
 the planes Q, §', §", we have the theorem: 
 
 The great circles lisecting the interior angles of a spherical 
 tria7igle meet in a 
 
 EXERCISES. 
 
 1. In order that the two planes P^P'= and P~P'— 
 may be perpendicular to each other, show that the coefficients 
 of X, y and ;2; in P and P' must satisfy the condition 
 
 2. Describe the relative position of the four planes 
 
 ^ + 2/ -1- 2 = 0, 
 
 x+ g-2z = 0, 
 X — 2g -\- z = 0, 
 
 and find the angles which each makes with the three others. 
 
THE PLANE. 259 
 
 3. Show that the line of intersection of the two planes 
 
 ax -\- hy -\- cz -\- d — 0, 
 ax -{-hy — cz -\- d — 0, 
 
 is in the plane of XY, and that its equation in this plane is 
 
 ax -\- hy -\- d = 0. 
 
 4. What is the condition that a plane shall pass through 
 the origin? 
 
 5. Write the equation of a plane making equal angles with 
 the three co-ordinate planes and cutting off from the axis of 
 JTan intercept a. 
 
 6. When a plane makes equal angles with the three co- 
 ordinate planes, what is the ratio of each inteicept which it 
 cuts off from the axes to the j)erpendicular from the origin 
 upon the plane? Ans. Vd : 1. 
 
 7. Write the equation of a plane which shall make equal 
 angles with the axes of Xand Z, and shall be parallel to the 
 axis of Y, 
 
 8. What is the distance apart of the parallel planes 
 
 X -f 2y + 2z = a; 
 2x -\- 4.y -{- 4rz = b? 
 
 9. Write the equation of the plane which shall pass 
 through the point (1, 2, 2) and be parallel to the plane 
 
 — x-^2y — z = 0. 
 
 10. Write the equation of the plane which shall pass 
 through the origin, the point (1, 1, 2) and the point (2, 3, 1). 
 
 A71S. — 6x -{- dy -{- z = 0. 
 
 11. Write the equation of the plane which shall pass 
 through the origin and the point (1, 2, 2), and shall be per- 
 pendicular to the plane 
 
 X — y -^ z = 0. 
 
 Ans. 4:X -f y — 3z= 0. 
 
 12. Find the locus of that point which is required to be 
 equally distant from the points (a, b, c) and (a', h', c'), 
 Ans. 2(a'- a)x + 2(6'- h)y + 2(c'- c)z 
 
 = a" - a' + b" -b' -\- c" - c\ 
 
260 GEOMETRY OF THESE DIMENSIONS. 
 
 13. If, in the preceding problem, the point {a', V , c') is 
 on the straight line from the origin to {a, b, c), and m times 
 as far from the origin as {a, h, c), show that the perpendicu- 
 
 m + 1 
 
 lar from the origin upon the plane is — - — Va" -\- b"^ -\- &. 
 
 14. The plane x -\- y -\- z — d = h required to bisect 
 the line from the origin to the point {a, l, c). Find the 
 yalue of d. Ans. d = ^a + Z* + c). 
 
 15. Find the equation of the plane passing through the 
 origin and through the line of intersection of the planes 
 
 2x + dy + 4:Z 4' 2^ = 0; 
 ^ + y + z — 22J = 0. 
 
 Ans. 6x -\- 7y -\- 9z = 0. 
 
 16. Find the equation of the plane which shall pass through 
 the point (2, 3, 5) and through the line of intersection of the 
 two planes 
 
 X -\- y -\- 2; — 5 = 0; 
 X — y-\-2z-{-l = 0. 
 
 A71S. X -{- dy — 11 = 0. 
 Calling the two expressions P and P', the equation, of any plane pass- 
 ing through the intersection of P and P' may be written in the form 
 XP-\-P' =0. We determine by the condition that this equation 
 shall be satisfied when we have x = 2, y = S and 2 = 5. 
 
 17. Write the equation of the plane passing through the 
 origin and perpendicular to the two planes 
 
 x-i-y- z = 0; 
 X — y — 2z = 0. 
 
 Ans. Zx — y -\-2z-= 0. 
 
 18. The three planes 
 
 X— 2y — 3z = 0, 
 2x-{- y — nz = 0, 
 Vx + m'y 4- 7i'z = 0, 
 are each to be perpendicular to the other two. Find the 
 least integral values of V, m', n' and n which satisfy this con- 
 dition, and thus show that the equations of the second and 
 third planes are 
 
 2a; + 2/ = 0; 
 Zx — ^ -\-hz=. 0. 
 
CHAPTER III. 
 THE STRAIGHT LINE IN SPACE. 
 
 243. Theokem I. The position of a line is completely 
 determined ly its projections tipon any two non-parallel 
 planes. 
 
 Proof. Through the projection on one of the planes pass 
 a plane ^ J_ to that of projection. The line projected then 
 must lie entirely in the plane R. 
 
 In the same way, the line must lie entirely in the plane 
 S Lio the other plane of projection and containing the other 
 projection. Hence the line is the intellection of the planes 
 R and S. 
 
 There can be only one plane R and one plane S, because 
 along a given line in a plane only one ± plane can be passed. 
 Hence there is but one line in which these planes can inter- 
 sect, and this is the line whose projections are given. Q. E. D. 
 
 244. Equations of a Straight Line. Since any one 
 equation between the co-ordinates of a point represents a sur- 
 face, at least two equations are necessary to represent a line 
 in space. These equations, considered separately, represent 
 two surfaces. Considered simultaneously, that is, requiring 
 the co-ordinates to satisfy them both, they represent the line 
 in which the surfaces intersect. 
 
 The most simple form of the equations of a straight line 
 are given by the equations of the planes in which it is pro- 
 jected upon any two of the co-ordinate planes, XZ and YZ 
 for example. The equation 
 
 X =^hz-\- a 
 
 (y being left indeterminate) represents a certain plane paral- 
 lel to the axis of Y{% 231); that is, the co-ordinates of all the 
 
262 GEOMETRY OF THREE DIMENSIONS. 
 
 points in this plane satisfy the equation, and vice versa. In 
 the same way, every point whose co-ordinates satisfy the equa- 
 tion 
 
 y z= Jcz -\- h 
 
 lies in a certain plane parallel to the axis of X. Hence 
 every point whose co-ordinates satisfy both equations must 
 lie in both planes, that is, in the line of intersection of the 
 planes. The two equations taken simultaneously therefore 
 represent a straight line. 
 
 Eemark. Any two consistent and independent simulta- 
 neous equations between the co-ordinates, for instance, 
 
 ax -^-hy -i-cz -\- d = 0, ) ,^. 
 
 a'x + I'y -f c'^ + ^' = 0, f ^ ^ 
 
 equally represent a straight line, namely, the line in which 
 the planes intersect. But the forms 
 
 . x = liz-\-aA /«v 
 
 y = hz^l,\ ('^ 
 
 are preferred because they are more simple. 
 
 We also remark that the form (1) can always be reduced 
 to the form (2) by first eliminating y and then x from the two 
 equations. 
 
 EXERCISES. 
 
 1. Express the equations of the line of intersection of the 
 planes 
 
 3a; - 2?/ + z^ hd = 0, 
 
 -x-{- y -^ 2z - 4:d = 0, 
 
 in the form (2). 
 
 2. Express in the form (2) the equations of the three lines 
 of intersection of the planes 
 
 X — y — z = a; 
 X -{- y — z = b; 
 
 ,x = - bz-Jr- 3d; 
 Ans. 
 
THE STRAIGHT LINE IN SPACE. 263 
 
 3. Explain how it is that the equation of a line in one of 
 the co-ordinate phines (the other co-ordinate being supposed 
 zero) is the same as tlie equation of the plane passing through 
 that line and parallel to the third co-ordinate. 
 
 4. Prove that if we represent the equations of a straight 
 line [(1) or (2), for example] in the form 
 
 P = 0, Q = 0, 
 then the equations 
 
 mP -\- nQ = 0, mP - nQ = 0, 
 771 and n being constants, will represent the same line. 
 
 245. Symmetrical Equations of a Straight Line. The 
 equations of a straight line may be represented, not only by 
 two equations between the three co-ordinates, but by express- 
 ing each of the three co-ordinates as a function of a fourth 
 variable. To do this, let us put 
 
 x^y ^o> ^o> ^^^^ co-ordinates of any fixed point of the line; 
 
 X, y, z, the co-ordinates of any other point of the line; 
 
 p, the length between the points {x^, y„, z^) and {x, y, z). 
 
 Then, a, § and y being the angles which the line makes 
 with the CO- ordinate axes, we have, by § 215, 
 
 a; — a;^ = p cos a\ \ 
 
 2/ - ^0 = P cos /?; V (3) 
 
 2 — ^^o = P cos /. ) 
 
 Here x^, y^, z^, a, ^ and y are supposed to be constants 
 which determine the position of the line in space, while 
 X, y, z and p are variables. Assigning any value we please to 
 p, we shall have corresponding values of x, y and z, which 
 will be the co-ordinates of that point P on the line which is 
 at the distance pfrom the point {x^, y^, z^). Since for every 
 point on the line there will be one and only one value of p, 
 and for this value of p one value and no more of each co- 
 ordinate, and vice versa, the equations (3) will represent all 
 points of the line, and no others. They are therefore the 
 equations of a straight line. 
 
 These equations (3) are readily reduced to the form (2) by 
 
264 OEOMETBY OF THREE DIMENSIONS. 
 
 eliminating p, first between the first and third, and then be- 
 tween the second and third. We thus find 
 
 cos a . x^ cos y — z. cos a 
 
 X = z + — ; 
 
 cos y cos y 
 
 cos /3 , y. cos V — z. cos /3 
 
 y = -z + — -. 
 
 ^ cosy cos y 
 
 The equations may also be reduced to the symmetric form 
 
 ^-^0 _ y-Vo _ 
 
 cos a cos /3 cos y 
 
 246. Introduction of Direction- Vectors. In the equa- 
 tions (3) we may introduce, instead of the direction-cosines, 
 any three quantities proportional to them, without changing 
 the line represented by the equation. Let these quantities be 
 I, m and n, so that the equations become 
 
 x = x^-\-lp; J 
 
 To show that the line is unchanged, we proceed as in § 217, 
 where we have shown that the proportionality of I, m and n 
 to the direction-cosines may be expressed by the equations 
 
 = (T cos a; 7n = ff cos /?; n = G cos y. 
 By substitution the equations (4) become 
 x = x^-\- p6 COS a\ 
 y^y.^-P^ cos /?; 
 z — z^-\- p<5 cos y. 
 
 These equations are the same as (3), except that pa takes 
 the place of p; that is, the distance between the points 
 (^o> ^0? ^o) ^^^ (^> y^ ^) is P^ instead of p. Hence the systems 
 (3) and (4) represent the same line, except that in (4) p repre- 
 sents length -^ a, instead of length simply. 
 
 Cor. We may multiply the three direction-vectors in the 
 symmetrical equations of a line hyany common factor without 
 changing the line represented. 
 

 
 ^^- ^<^^^ ^-^/--^-^^ 
 
 Jfu-r^^ ^IlZJI' y.-'/^ -z _2 __j? 
 
 
THE STRAIGHT LINE IN SPACE. 265 
 
 Remark. The forms (3) and (4) have a great advantage 
 in nearly all the investigations of Analytic Geometry, and 
 will therefore be exclusively employed. The advantage 
 arises from the fact that the three co-ordinates which fix the 
 position of some one point of the line are completely distinct 
 from the quantities I, m and n which express its direction. 
 
 EXERCISES. 
 
 1. Express the co-ordinates of the three points in which 
 the line given by the equations (3) intersects the three co- 
 ordinate planes respectively. Express also the corresponding 
 values of p. 
 
 2. Write, in the form (4), the equations of a line passing 
 through the point (a, h, c) and parallel to the axis of Z. 
 
 3. Write, in the same form, the equations of a line passing 
 through the point {x^, y^, z^ parallel to the plane of XJTand 
 making equal angles with the axes of Xand Y. 
 
 4. Write the equations of a line passing through the point 
 (x^, y^, z^ and making equal angles with the co-ordinate 
 planes. Express also tlie co-ordinates of the three points in 
 which it intersects the co-ordinate planes. 
 
 Ans., in part. It intersects the plane of YZ in the points 
 
 y = yo - ^o'^ ^ = ^0 - ^0- 
 
 5. Show that the equations of the line passing through the 
 points {x^y y^f z^) and (x^, y^, z^) may be written in the form 
 
 X = x^ -\- (x^ — xjp; f>^^-< <^y<'^ ts /^^ ,^^ 
 
 y^y.^{y.- y.)P', ^r'^'T- T'^ti^ 
 
 state to what distance on the line corresponds the unit of p 
 in these equations, and find the co-ordinates of the points in 
 which the line intersects the co-ordinate planes. 
 
 6. Write the three symmetrical equations of the straight 
 line joining the points (1, 1, 2) and (2, 3, 5). Find the 
 angles which it makes with the co-ordinate axes, tlie points 
 in which it intersects the co-ordinate planes, and the distances 
 between these points. 
 
266 GEOMETRY OF THREE DIMENSIONS. 
 
 347. Condition that a Line shall he ])arallcl to a Plane. 
 So long as the coefficients I, m and n in the equations (4) of 
 a straight line are entirely unrestricted, these equations may, 
 by giving suitable values to I, m and n, be made to represent 
 any line whatever passing through the point (x^, y^y 2; J. If, 
 however, they be subjected to a homogeneous equation of con- 
 dition, the lines will be restricted, as we shall now show. 
 
 Theorem II. If, in the symmetrical equations of a 
 straight line, the direction-vectors m, n and fp are required 
 to satisfy a linear equation, the line will lie in or he j^nrallel 
 to a certain inlane. 
 
 Conversely, the requirement that the line shall lie in or he 
 parallel to a certain pla7ie is indicated hy a linear equation 
 hetioeen the direction-vectors. 
 
 Proof. Let the linear equation which w, n and p are re- 
 quired to satisfy be 
 
 Al + Bm -f Cn = 0. {a) 
 
 I say that every point of every possible line represented by the 
 equations (4) will then lie in the ^olane whose equation is 
 
 Aix - X,) + B(y - y,) + C(z - z,) = 0, (h) 
 
 and will therefore be parallel to every plane whose direction- 
 vectors are A, B and C. For, by multiplying the equations 
 (4) respectively by ^, -S and G, transposing, and adding the 
 products, we find 
 
 A{x - X,) + B{y - y,) + C(z - z,) = (Al + Bm + Cn)p. 
 
 I^ow, by hypothesis (a), the second member of this equation 
 vanishes. Hence all values of the co-ordinates x, y and z 
 which satisfy (4) also satisfy [h). Hence every point of the 
 line lies in the surface whose equation is (h), and this surface 
 is a plane, by § 229. 
 
 Every plane whose direction-vectors are ^, B and G is 
 parallel to (h), because perpendicular to the same line. Hence 
 (a) is the condition that the line (4) is parallel to every such 
 plane. 
 
THE STRAIGHT LINE IN SPACE. 267 
 
 Next, let it be required tluit the line (5) shall lie in the 
 plane whose equation is 
 
 Ax + Bi/ -^ Cz -{-D= 0. (c) 
 
 I say that the coeflBcients m, n and f must satisfy the linear X 
 equation ^ 
 
 Al + Bm + Cn = 0. 
 
 For, by substituting in (c) the values of x, y and z from (4), 
 we have 
 
 Ax, + By, + Cz, + D + (M + Bm + Ch)p = 0, {d) 
 
 which equation must be satisfied for all values of p. Now, by 
 hypothesis, the point {x^, y^, z^ lies on the line, and therefore 
 lies in the plane (c) which requires it to satisfy the equation 
 
 -4^-0 + %o -^Cz,-\-D = Q. 
 
 Hence, in order that the equation {d) may be satisfied, we 
 must have 
 
 Al + Bm + Cn = 0. (5) 
 
 248. Common Perpendicular to Two Lines. It is shown 
 in Geometry that two non-parallel lines have one and only one 
 common perpendicular, and that this perpendicular is the 
 shortest distance between the lines. Let us now solve the 
 problem, 
 
 To find the equation of the common perpendicular to two 
 given lines. 
 
 We shall express the equations of the given lines in the 
 form (3), putting, for brevity, 
 
 ^u Pv Yv I the direction-cosines of the ffiven lines, 
 ^.> A. r.. ' 
 and 
 
 a, /?, ;/, those of the required perpendicular. 
 Thus the symmetrical equations of the given lines will be 
 x = x^-\- a^p', \ rx = x^-\- a^p\ 
 
 y = yr-\- P^9\ \ and j 7/ = y^ -I- ^^p; 
 
 ^ = ^» + r,P; ' Kz ^ z,A,- y^p. 
 
268 
 
 OEOMETRT OF THREE DIMENSIONS. 
 
 Let as first find the direction-cosines a, p, y. By §§ 216, 
 217, we have the equations 
 
 a^a -f fi^p + y^y = 0; 
 
 (6) 
 
 Eliminating first yS and then /from these equations, we have 
 
 (a^p, - a^P^)a + (P^y^ - p^y^)y = 0; 
 {y,a, - y^a^)a + (/3,y^ - /?^;/J/? = 0. 
 
 Dividing these equations by ay and a/3, respectively, gives 
 
 or /? ~ r ~ 
 
 /^^ = Ar. -AKi; 
 
 Mr = «^iA - «^2;^i- 
 
 Taking the sum of the squares of these equations, 
 
 M' = (Ar, - ArJ' + (r.«. - n^^Y + («.A - «,A)^ (7) 
 
 which is the square of the sine of the angle between the given 
 lines (§ 218). 
 
 The direction-cosines a, /? and y are therefore 
 
 a = —^-^ 
 
 Ar,. 
 
 sin V 
 
 /3=y^ 
 
 n^^. 
 
 sm V 
 
 sin ^' 
 
 (8) 
 
 t^ being the angle between the given lines. Thus the direc- 
 tion of the required line is completely determined. 
 
 To complete the solution, we must find the co-ordinates 
 of some point of the line. Let us then pat 
 
 (a, b, c) the point in which the required line intersects the 
 first of the given lines. The equations of the required line 
 may then be written 
 
THE STRAIGHT LINE IN SPACE. 269 
 
 x= a -{- ap; \ 
 
 z =c-\-yp.) 
 
 Let us also put 
 
 p„ the distance of the point (a, by c) from (x^, ?/,, z^) on 
 the first given line ; 
 
 p^, the distance from {x^, y^, z^) on the second given line 
 to the point in which the required line intersects it; 
 
 Po, the distance of the points of intersection, that is, the 
 length of the shortest line between the given lines. 
 
 Then, equating the expressions for the co-ordinates of the 
 points of intersection on the two lines, we have the six equa- 
 tions 
 
 a = x^-\- «iPi; ) Intersection of required 
 b =2/1+ APiJ ( ^ii^6 with first given 
 c = 2;, -f y^p^. ) line. 
 
 a -f- ^Po = ^2 + ^iPil ) Intersection of required 
 b -\- /?Po = 2/2 + A/^aJ f ^^^^ wi^^ second given 
 c -\- ypo = ^, -\- r,P,' ^ line. 
 
 (c) 
 
 These six equations suffice to completely determine the six 
 unknown quantities, a, b, c, p^, p,, p^. First subtracting 
 corresponding equations in the two sets, we eliminate a, b and 
 c, and have three equations which we may write in the form 
 
 ^Po + ^iPl - ^2P. = ^1 - ^,» ) 
 
 M + ^^p^ - Ap. = y.- y^^ \ W 
 
 yPo + r^p^ - r,p, = ^, - ^^, ' 
 
 and which contain only the three unknown quantities p^, p^ 
 and Pj. Multiplying the equations in order by a, /? and y, 
 taking their sum and referring to the relations (6), we have 
 
 po = ^K - ^i) + ^(y. - y,) + rfe - ^J- (9) 
 
 From the manner in which p^ has been defined, it is equal 
 to the shortest distance between the two lines (1) and (2), be- 
 cause it is the distance from the point [a, b, c) to the point 
 in which the shortest line intersects the second line. 
 
270 GEOMETRY OF THREE DIMENSIONS. 
 
 If we substitute for or, /? and y their values (8), we have 
 
 
 (10) 
 
 Again, multiplying the equations (d) by a„ y^i and ;^i, and 
 adding, and then by a^, /?, and ;/„ and adding, we find 
 
 Pr - P. cos V = a^x^ - x^ 4- /?,(?/, - ?/J 4- ;/X2^2 - ^i) = ^; 
 /9, cos V - p, = a,(a:, - a:,) + /?,(y, - ?/J + ;/,(2;, - z^ = r,. 
 
 Hence 
 
 _ ^\ ~ '^\ cos V 
 ^' ~ sm^ ' 
 
 r, cos V — r. 
 
 P^ = - — —2 ^• 
 
 ^ sin V 
 
 To find a, b and c, we have only to substitute the value of 
 p, in (c), which gives 
 
 , a,r, — ar^ cos v 
 
 a = .T, + --5-5^ — H-^ ; 
 
 ' sin V 
 
 ^ ~ ^^ "^ sin' V 
 
 . K,^', — Ki^o cos V 
 
 C z=z Z. -\- ^-^ /-f-^ . 
 
 ' sin^ V 
 
 (11) 
 
 The values of a, Z>, c, a, y^and y in (11) and (8) being 
 substituted in (Z>), the equations of the shortest line are com- 
 plete. 
 
 249. Condition of Intersection. Since p^ in (9) expresses 
 the shortest distance between the two lines, the condition 
 that the lines shall intersect is found by putting p — 0. 
 Substituting for a, ft and y their values (8), this condition 
 gives 
 
 If, instead of or, ft and y, we use the quantities Z, m and n, 
 we must, from the proportionality of these factors to a, ft and 
 )/, have a, ft and )/ equal respectively to /, m and w, each 
 multiplied by the same factor. 
 
THE STRAIGHT LINE IN SPACE. 271 
 
 If we call p and q these factors in the cases of or,, /?„ y^ 
 and «,, /?„ ;^, respectively, we have 
 
 and the condition of intersection becomes 
 
 (w,7la—Wani)(ira-a'i)+(;iiZ2 — 712^1) (ya-2^i)-f(ZiWa-^2W,) (2a— Si)=0. (12) 
 
 250. Problem. To find the point in which a lirie m- 
 tersects a surface. 
 
 Since the point of intersection lies on the line, there will 
 be a definite value of p corresponding to it. This value of p, 
 being substituted in the equation of the line, will give values 
 of the co-ordinates x, y and z which, if p is properly taken, 
 will satisfy the equation of the surface. We therefore proceed 
 as follows: 
 
 Calling, for the moment, {a, b, c) any one point of the 
 given line, we substitute in the equation of the surface, for 
 X, y and z, the expressions 
 
 X = a -\- Ip', y =zh -\- mp; z = c -{- np. {a) 
 
 The equation of the surface will then contain no unknown 
 quantity except p, and is to be solved so as to get an expres- 
 sion for p which shall satisfy it. 
 
 This expression being substituted in the equations {a) 
 will give the required values of the co-ordinates of the point 
 of intersection. 
 
 If the equation in p is of a higher degree than the first, 
 there will be several values of p, and therefore several points 
 of intersection. 
 
 Example. Find the point in which the line 
 
 a; = 2 + 2p, 
 3/ = 3 - 2p, 
 2; = 5 — p, 
 intersects the plane 
 
 2a; - 3y - 2J -f 8 = 0. 
 
272 OEOMETRT OF THREE DIMENSIONS. 
 
 Substituting the values of the co-ordinates in the equa- 
 tion of the plane, we find 
 
 which gives P — T\ 
 
 10 + lip + 8 = 0, 
 2 
 11' 
 
 whence 
 
 ^ = 2A> y = 2t\, z = 4^, 
 
 are the co ordinates of the point of intersection. 
 
 The same general method applies whenever points fulfil- 
 ling any condition whatever are to be found on one or more 
 lines. Each line must have its own value of p, which we 
 may distinguish from the values for other lines by accents or 
 subscript numbers. The values of the co-ordinates, expressed 
 in terms of p, are to be substituted in each condition, and 
 equations with the p's as the only unknown quantities will 
 thus be formed. 
 
 EXERCISES. 
 
 1. Write the equations of the sides of the triangle whose 
 vertices are at the points (1, 2, 3), (3, 2, 1) and (2, 3, 1), and 
 find the angles of the triangle. 
 
 Ans., 171 part. 30°, 60°, 90°. 
 
 2. Find the points in which the line joining the points 
 (1, 2, 3) and (2, 3, 4) intersects the co-ordinate 23lanes. 
 
 Ans. (0, 1, 2); (- 1, 0, 1); (- 2, - 1, 0). 
 
 3. Write the symmetrical equations of the line passing 
 through the point (a, b, c) and perpendicular to the plane 
 
 px -{- qy -\- rz = 0. 
 
 4. An equilateral triangle has one vertex in each co-ordi- 
 nate plane, at the distance h from each of the axes lying in 
 that plane. Write the equations of each of its sides, taking 
 the middle point of each side as the point from which p is 
 measured. 
 
 Ans., in part, x = h; ) Equations 
 
 y =:^h -\- p;y of one of 
 z =ih — p.) the sides. 
 
THE STRAIGHT LINE IN SPACE. 273 
 
 5. Ill what points does the line of intersection of the two 
 planes 
 
 x-{-ij -z ^7, 
 x-y-\-2z = l, 
 
 intersect the co-ordinate planes? 
 
 Ans. (4,3,0); (0,15,8); (5,0,-2). 
 
 6. Express the point in which the line (4), § 246, inter- 
 sects the plane Lx -f- My -{- Nz = 0. 
 
 A ' i M{inx^ — ly^) + N(nx^ — Iz,) 
 
 Ans., m part, x = — ^^ — V v . \ r — , ,x -• 
 
 ^ lL-\- Mm + Nn 
 
 7. Write the symmetrical equations of the line of inter- 
 section of the two planes 
 
 a; + 2?/ — 30 — 5 = 0, 
 2a;-?/ +22^ + 7 = 0, 
 
 taking as the zero point of the line that in which it intersects 
 the plane XY. 
 
 9 17 
 
 Ans. X = — —— p', y z=—- -\- 8p; z = -{- 6p. 
 
 O 
 
 In cases where direction-vectors appear as unknown quantities in 
 equations, there will be but two equations for the three vectors. In this 
 case we determine any two in terms of the third, and assign to the lat- 
 ter such value as will give the simplest form to the results. 
 
 8. Write the equations of the line passing through the 
 point (3, 1, 5) and intersecting the axis of X perpendicularly. 
 
 9. Write the equations of the line passing through the 
 point {a, l, c) and parallel to each of the planes 
 
 x-\-y-^z = 0', 
 X — y -\- z =0. 
 
 Ans. X = a -{- p; 
 y = h-{-Sp; 
 z = c -{- 2p. 
 
 10. Find the condition that the line (4), § 246, shall inter- 
 sect the axis of Z. Ans. mx^ = ly^. 
 
 The condition requires that the points in which the line intersects the 
 planes of XZ and YZ respectively shall be the same, that is, correspond 
 to the same value of p. 
 
274 GEOMETRY OF TEHEE BIMEJS8I0KS. 
 
 11. Find the equation of the line passing through the 
 point (5, 2, 4) and intersecting perpendicularly that line 
 through the origin whose direction-vectors are I = 1, m = 2, 
 n = 2. Ans. x = 5 — 14p; y = 2 -\- 8/3; z = 4: — p. 
 
 Write the symmetrical equations of the given and the required line, 
 calling p' the variable for the one line, and p for the other. The con- 
 dition that some one point {x, y, z) shall satisfy the equations of both 
 lines then gives three equations of condition between I, m, n, p' and p, 
 and the condition of perpendicularity gives a fourth, 
 
 12. Deduce the condition that two lines shall intersect by 
 the principle that there must then be 07ie set of values of x, y 
 and z which shall satisfy the equations of both lines, these 
 values of the co-ordinates being given in terms of one value 
 of p for the one line, and another value for the other line. 
 
 The condition gives three equations between the two quantities p 
 (on one line) and p' (on the other), and the values of p and p' must be 
 the same, whether we derive them from one or another pair of the equa- 
 tions. 
 
 13. Write the equation of the plane which contains the 
 two intersecting lines 
 
 x = a-]- p\ 
 
 x = a- p; 
 
 y=zh-p; 
 
 y = d-2p; 
 
 z =c-2p; 
 
 z =c -3p. 
 
 A71S. X — 6y -\- Sz — a -\- 5b — 3c = 0, 
 
 Note that the condition that a plane shall contain or be parallel to a 
 line is the same as that a line shall be parallel to or lie in a plane. 
 
 14. Find that plane which is parallel to each of the lines 
 
 X = a — 2p; X = a^ -\- p; 
 
 y = b-p\ y=zb'^2p', 
 
 z = c -\- p; z = c' — p; 
 
 and equidistant from them. Also find the common distance. 
 A?is. 2x -\- 2y -\- 6z — a - a' — b — y — Sc — 3c' = 0. 
 
 Common dist. = ^ -«' + ^ - ^1+ 3(c - O 
 
 2i/ll 
 
CHAPTER IV. 
 
 QUADRIC SURFACES. 
 
 General Properties of Quadrics. 
 
 251. Def. A quadric surface is the locus of a point 
 in space whose co-ordinates are required to satisfy an equation 
 of the second degree. 
 
 Remark. A quadric surface is called a quadric simply. 
 The most general form of an equation of the second degree 
 between the co-ordinates is 
 
 gx^ + hy^ + hz" + 2g'yz + Wzx + Wxy 
 
 + 2g"x + W'y + Wz -\-d = Q, (1) 
 
 because the terms in this equation include all powers and 
 products of the co-ordinates x, y and z up to those of the 
 second degree. 
 
 The number of coefficients, as written, is ten. But since, 
 by division, we may reduce any coefficient to unity, their 
 number is, in effect, equivalent to nine. Hence: 
 
 Theorem I. Nine conditions are necessary to determine 
 a quadric in space. 
 
 Eemark. In discussing the equation (1), we regard the 
 coefficients^, h, h, g' , etc., as given constants, unless other- 
 wise expressed. 
 
 We may trace out certain analogies between the quadric 
 and conic, by treating tlie former in the same general manner 
 in which we treated the latter in Part I. 
 
276 OEOMETRT OF THREE DIMENSIONS. 
 
 252. hiter sections of a Q^iadric with a Straight Line, 
 Let the equations of the line be 
 
 ^ = ^0 + ^pn 
 
 y = yo + ^^ip; r («) 
 
 Z = Z^-i- 7ip. ) 
 
 The problem now is, What values of x, y and z satisfy both 
 these equations and (1)? (Of. § 250.) If we substitute these 
 expressions for x, y and z in (1), thus: 
 
 gx'^=g(x:^nx,p^rfy^), 
 etc. etc., 
 
 we shall have an equation in which all the quantities except 
 p will be supposed given. Hence p can be determined from 
 the equation. When this is done, the values of the co-ordi- 
 nates of the point or points of intersection are found by sub- 
 stituting the values of p in {a). Now the equation in p will 
 be of the second degree, and will therefore have two roots, 
 which may be real, imaginary or equal. Hence: 
 
 Theokem II. Every straight line intersects a quadric in 
 two real, imaginary or coincident poi^its. 
 
 253. Centre of Quadric. Let us next change the origin 
 to a point whose co-ordinates have the arbitrary values A, B, 
 G, If we distinguish the co-ordinates referred to the new 
 system by accents, we shall have (§219) 
 
 X ^ x' -\- A\ 
 y=y'-\-B', 
 z = z' + 0, 
 
 Substituting these values in the general equation (1), it be- 
 comes 
 
 gx'"" + 7iv" + hz" + 2g'y'z' + Wz'x' + Wx'y' 
 + Kg A H- k'B + h'C + g")x' 
 + KhB J^g'C^k'A-{- h")y' 
 -f-2(^•(7-f A'^ -\-g'B + k")z' 
 + gA' + hB' ^TcC + %g'BC^ WOA + WAB 
 
 + 2^"^ + WB -f U" O-\-d = 0. (1') 
 
qUADRIC SURFACES. 211 
 
 Let us now determine the co-ordinates A, B, C ot the new 
 origin by the condition that the coefficients of x% y' and z' 
 shall all vanish. To do this we have to solve the equations 
 
 gA J^lc'B^li'C= -cj";\ 
 
 h'A -\-hB -{-g'C ^ - //'; \ (2) 
 
 ¥A + g'B + y^C = _ Ic"; ) 
 
 regarding A, B and G as the unknown quantities. Since 
 there are as many equations as unknown quantities, the solu- 
 tion will, in general, be possible. 
 
 Let us now suppose the equations (2) to be solved, and the 
 resulting values of A, B and C to be substituted in (1'). Let 
 us also put 
 
 d! , the absolute terms in (1'). 
 
 Then, omitting accents, the equation (1') of the quadric re- 
 duces to 
 
 gx^ 4- Tiy" + hz^ + ^'yz + %li'zx + %¥xy + ^' = 0. (3) 
 
 From this equation we may deduce a second fundamental 
 property of the quadric. If {x, y, z) be any values of the co- 
 ordinates which satisfy (3), it is evident that (— x, — y, — z) 
 will also satisfy it. Hence, if one of these points is on the 
 quadric, the other will also be on it. But the line joining 
 these points passes through the origin, and is bisected by the 
 origin, that is, by the point whose co-ordinates, referred to the 
 original system, are (A, B, C). Since {x, y, z) may be any 
 point on the quadric, we conclude: 
 
 Theorem III. For every quadric there is, tn general, a 
 point loliicli bisects every chord passing through it. 
 
 Def. That point which bisects every chord passing 
 through it is called the centre of the quadric. 
 
 A chord through the centre is called a diameter of the 
 quadric. 
 
 354. Section of a Quadric hy a Plane. To investigate 
 the equation of the plane curve in which any plane intersects 
 the quadric, we may take a pair of co-ordinate axes in the 
 cutting plane. This we do by simply transforming the equa- 
 tion to one referred to new axes; and we may, in the first 
 
278 OEOMETRT OF THREE DIMENSIONS. 
 
 place, leave the origin unchanged. Accenting the new co- 
 ordinates, the equations of transformation will be (§220) 
 
 x = ax' -f Py' 4- yz'-, 
 y = a'x' + ^'y' + y'z'-, 
 % = a'V + y3"y' + ;/'V; 
 
 ^f A Vf etc., being the direction-cosines of the new co-ordi- 
 nate axes relatively to the old ones. 
 
 Now when we substitute these expressions in the general 
 equation (1) and arrange the terms according to the powers 
 and products of x', y' and z', we shall have a new equation of 
 the same form as (1), that is, one containing terms in a:", 
 y'^y /% x'y', etc. ; the only change being that the coefiScients 
 gy h, h, g'y etc., have new values. We may therefore, without 
 any loss of generality, take the equation (1) as representing 
 the transformed equation, and consider the section of the sur- 
 face which it represents by a plane parallel to any one of the 
 co-ordinate planes, XY for example. Let us then suppose 
 z= c m (1). The equation of the section of the quadric by 
 the plane z = c will be, omitting the accents, 
 
 gx' + hf + Wxy -f 2{h'c + g")x + ^g'c + h")y 
 
 + Tcc^ + Wc + J = 0. 
 
 This is the equation of a conic section. Hence: 
 
 Theorem IV. Every plane section of a quadric is a conic 
 section. 
 
 It is algt) shown in § 198 that all conies whose equations 
 have the same coefficients in x^, xy and y"^ are similar and 
 similarly placed. Now, in the above equation, the coefficients 
 g, li and 2^ remain unaltered, however c may change; that is, 
 however we may change the position of the cutting plane, so 
 long as it remains parallel to the plane of XY. Hence: 
 
 Theorem. V. All sections of a quadric ly parallel 
 planes are similar conies and have their principal axes 
 parallel. 
 
 Cor. If any plane section of a quadric is a circle, all 
 sections parallel to it are circles. 
 
QUADBIC SURFACES. 279 
 
 EXERCISES. 
 
 1. Find the centre of the quadric 
 
 x^ + /-^^^ + 2;' + nyz -\- mx = 0. 
 
 2. Write the equation of the locus of the point required 
 to be equally distant from the origin and from the plane 
 
 ax-\:fiy^yz-p^O. K + ^^ + r^ = l.) §^~ 
 
 3. Write the equation of the locus of the point equally 
 distant from the origin and from tlie plane 
 
 CX-\-c'tJ -p= 0, (6'^ + c'' = 1.) 
 
 and show that its centre is at infinity. 
 
 255. Conjugate Axes and Planes. Consider this prob- 
 lem: 
 
 To find the locus of the middle points of all chords of a, 
 quadric parallel to any fixed line, and therefore to each other. 
 
 Let the equation of any one of the chords be 
 
 X = x^ + Ip', 
 
 y = !/o + ^^p; 
 
 z = z^ -^np. 
 
 If I, m and n remain constant, then, by asigning all values to 
 x^, y^ andz^, these equations may represent any system of lines 
 parallel to each other. Now, we find the two points in which 
 any one of these lines intersects the surface by the process of 
 § 250; namely, we put in the equation (3) of the surface 
 
 x'^ = x,' + 2lx,p-^rp^;. 
 
 y' = y' + 2^^2/oP + '?^>'; 
 
 z" =zj' -\-27iz^p + n'p'; 
 
 y^ = yo^o + (^^^0 + '^^^o)P + tnup""', 
 zx = z^x^ 4- {nx^ + lz^)p + nlp^', 
 
 ^y = ^,y. + (^2/0 + ^^o)p + i'nip\ 
 
 For brevity, let us represent the result of substituting these 
 values in (3) in the form 
 
 A^Bp-{- Cp' = 0. (a) 
 
280 GEOMETRY OF THREE DIMENSIONS. 
 
 Now, we may choose for (a;„, y^, z^) any point on the chord. 
 Let us choose the middle point. This point will be deter- 
 mined by the condition that the two values of p from the 
 quadratic equation {a) shall be equal, with opposite algebraic 
 signs. The condition for this result is ^ = 0. That is, writ- 
 ing for B its value, the condition will be 
 
 glx, + hmy, + knz, + g^ny, + 7nz,) + h'(7ix, + Iz,) 
 
 + k\ly, + mx,) = 0. 
 
 This, then, is the equation which the middle point {x^, y^, z^) 
 must satisfy as x^, y^ and z^ vary. Being of the first degree, 
 it is the equation of a plane, and, having no absolute term, 
 the plane passes through the origin, that is, the centre 
 of the quadric. Hence: 
 
 Theorem. VI. The locus of the 7nicldle points of a system 
 of parallel chords of a quadric is a plane through the centre. 
 
 Def A plane through the,centre of a quadric is called a 
 diametral plane. 
 
 That diametral plane which bisects all chords parallel to a 
 diameter is said to be conjugate to such diameter, and the 
 diameter is conjugate to the plane. 
 
 That diameter whose direction-vectors are I, m, n, that is, 
 whose equations are 
 
 X = Ip, 
 y — mf>, 
 z = np, 
 
 will be called the diameter {}, m, n). 
 
 Remark. If we call any diameter ^, we may call the 
 conjugate diametral plane A'. 
 
 The above equation of the diametral plane may be written 
 out thus, the subscript zeros being omitted: 
 
 {gl -j- ¥m + ¥n)x + {¥1 + hm -}- g'7i)y 
 
 + {hn 4- g'm + hn)z = 0. (4) 
 That is, this plan^ is conjugate to the diameter {I, m, n). 
 
 Theorem. VI J. If a diameter B lie in a plane A\ the 
 conjugate diametral plane B' will contain the diameter A, 
 conjugate to A'. 
 
QUADRIC SURFACES. 281 
 
 Proof. Let the equation (4) represent the diametral 
 plane A', and let the diameter B be (A, //, v). By § 24:7, tlie 
 condition that this diameter shall lie in the plane (4) is 
 
 {gl -f h'm -\-h'n)X + {k'l -f lim + g'n)}j. 
 
 + {h'l + g'm + /(;?0^ = 0, 
 or, rearranging the terms, 
 
 {gX 4- F// + ^V)? 4- {k'X + 7i// + ^V)77i 
 
 + {h'X 4- ^'yw + ky)n = 0. 
 
 But (§ 247) this is the condition that the diameter (/, m, n), 
 or A, shall lie in the plane 
 
 (# + ^V + h'r)x + (FA + hpi + ^V)?/ 
 
 which, by comparison with (4), is seen to represent the plane 
 conjugate to the diameter (A, yw, r), or B; that is, the plane 
 B\ Hence this plane contains the diameter A. Q. E. D. 
 
 Scholium. Having two conjugate diameters, A and By 
 with their diametral planes. A' and B', arranged as in this 
 theorem, the intersection of the planes A' and B' will deter- 
 mine a third diameter, which we may call C. Then, because 
 C lies in both the planes A' and B', its conjugate plane (7' 
 will, by Theorem VII., pass through both A and B. Thus 
 we shall have a system of three diametral planes whose inter- 
 sections will form three diameters, and each plane will bisect 
 all chords parallel to its conjugate diameter. 
 
 These three lines and planes are called a system of con- 
 jugate axes and diametral planes. 
 
 2^Q, Change in the Direction of the Axes. To simplify 
 the equation (3) still further, let us change the direction of 
 the axes of co-ordinates, leaving the origin at the centre. 
 This we do by the substitution 
 
 X = ax' 4- fty' 4" yz'', 
 y = a'x' 4- /3'y' -\- y'z*\ 
 z = a'V 4- /?"?/' 4- y''z\ 
 
 If we substitute these values in (3), we shall have an equation 
 
282 GEOMETRY OF THREE DIMENSIONS. 
 
 the terms of wliicli we can arrange according to the powers 
 and products of x', y' and 2'; namely, 
 
 ^'\ y'\ z'\ y'z', z'x\ x'y\ 
 We then suppose the values of the direction-cosines a, /?, y^ a\ 
 etc., to be so determined that tlie coefficients of y*7J , z*x' and 
 x'y' shall all three vanish. This will require three equations 
 of condition to be satisfied, which, with the six relations (14) 
 of §220, will completely determine the nine direction-cosines.* 
 These cosines being determined, the coefficients of a;"*, y'"^ 
 and z'"^ will all become known quantities, while the products 
 y*z', etc., will disappear. Thus, omitting once more the 
 accents from the co-ordinates, the equation (3) will be re- 
 duced to the form 
 
 Vx"" -\-m'y- ■\- n'z" ^ cV ^^, 
 
 -y-ff-p = l, (5) 
 
 V, w' and 71' being known quantities, functions of the origi- 
 nal coefficients in (1). It will be seen that the absolute term 
 d^ remains unaltered by this transformation. 
 
 The several quantities -7,, -r,-, etc., may be either positive 
 
 or negative, according to the values of the coefficients wiiich 
 enter into the original equation (1). 
 
 257. Principal Axes. Let us put A for the value of 
 
 d' . ^' 
 
 -,- taken positively; the first term of (5) will be ± -:, accord- 
 
 ing to whether it is positive or negative. If then we put 
 a = VI=V± J, 
 
 the term will become ± -..- 
 a' 
 
 * The equations obtained in this way are too complex for con- 
 venient management, and the actual values of the direction-cosines 
 must be found by the differential calculus, or by an application of the 
 algebra of linear substitutions. "We must therefore, at present, be con- 
 tented with showing the possibility of the solution, which is all that is 
 necessary for our immediate purpose. 
 
QUAD RIG SURFACES. 283 
 
 In the same way the other terms can be rcdnced to the 
 form ± 'jif and ± -, . Thus the general equation of the quad- 
 
 
 
 ric can finally be reduced to the form 
 
 ±~±--fr±-, = l. (6) 
 
 a b- & ^ ' 
 
 Def. The quantities a, h and c in this equation are called 
 the principal axes of the quadric. 
 
 The Three Classes of Quaclrics. 
 
 258. There are now four possible cases, omitting the 
 exceptional ones in which a, h, or c is zero or infinity. 
 
 Case I. The coefficients of the first member of (6) all 
 positive. 
 
 Case II. Two coefficients positive and one negative. 
 
 Case III. One coefficient positive and two negative. 
 
 Case IV. All the coefficients negative. 
 
 In the last case no real values of the co-ordinates can sat- 
 isfy the equation, because the terms, being themselves squares, 
 are essentially positive, and therefore with the minus signs 
 essentially negative. Hence there can be no real surface to 
 represent the equation. But in the other three cases there 
 will be real loci. Hence 
 
 Tliere are three rjeneral classes of real quadrics. 
 
 259, Class I. The EUiiJsoid. In Case I. the equation 
 is 
 
 -- 4- ^- + -- = 1 (7) 
 
 Let us first investigate the limiting values of the co-ordi- 
 nates. "Writing the equation in the form 
 
 ^ I l' - 1 _ 5 
 
 we see that when — c > 2; > -\-^c, the co-ordinates x audi/ can- 
 not both be real. Hence the surface is wholly included between 
 the two planes ^ = -f ^ a^'^d z =. — c. 
 
284 
 
 GEOMETRY OF THREE DIMENSIONS. 
 
 In the same way it is shown that the surface is included 
 between tlie planes x =^ -\- a and x — — a, and also between 
 the planes y — -\-h and y =i —h. Hence it is bounded in 
 every direction. 
 
 Because its sections by a plane are of the second order, 
 and limited in extent, they must all be ellipses. Hence the 
 surface is called an ellipsoid. 
 
 If we suppose z = ± c, we have x = and ?/ = 0, as tlio 
 only values of x and y which can satisfy the equation. Hence 
 each of the two planes z = -\- c and z = — c meets the sur- 
 face in a single point on the axis of Z, and is therefore tan- 
 gent to the surface. Extending the same proof to the other 
 two co-ordinates, we reach the conclusion: 
 
 The e^kt planes x = -\- a, x = —a, y = -{-h, y = — h, 
 z = -{- c and z = — c are all tajigeiits to the ellipsoid at the 
 points which lie on the axes at the distances ± a^ ±b and ± c 
 from the origin. 
 
 These six planes form the faces of a rectangular parallelo- 
 piped whose edges are respectively %a, 2b and 2c. Each pair 
 of parallel faces being at equal distances on the two sides of 
 the origin, and parallel to the corresponding axes, these axes 
 intersect the faces in their centres. Hence: 
 
 Theorem VIII. Every ellipsoid may be inscribed in a 
 rectajigular parallelojjiped whose surface it will touch in the 
 centre of each face. 
 
 260. Class II. The Hyperboloid of One Nappe. Let us 
 take that form of the equation (7) in which one of the three 
 
QUADBIG SURFACES. 285 
 
 terms of the first member is negative. Suppose this term to 
 be that in z. The equation is then 
 
 ^V ^: _ !! - 1 (8) 
 
 which we may write in the form 
 
 3 2 2 
 
 l + F = ^ + ?- . (^') 
 
 Let us now find the curve in which the surface intersects 
 the plane of XY. This we do by putting 2 = 0, which gives 
 at once the equation of an ellipse whose major axes are a and 
 h. Hence: 
 
 Theorem IX. The hyperholoid of one nappe intersects 
 the plane of XY in an ellipse whose axes are the same as the 
 axes' a and I of the surface. 
 
 This ellipse is called the ellipse of the gorge. 
 
 Let us next find the curve in which the surface intersects 
 a plane parallel to that of XY and at a distance k from it. 
 The equation of such a plane is 
 
 z = k. 
 Substituting this constant value of z, and putting, for brevity, 
 
 V = l + %, (a) 
 
 c 
 
 the equation (8') reduces to 
 
 This is the equation of an ellipse whose axes are ha and 
 hh. Whatever the value of h, the ratio of these axes will be 
 a : h, so that the ellipses will be similar. Hence: 
 
 Theorem X. The hyperloloid of one nappe cuts all planes 
 perpendicular to its axis of Z in similar ellipses. 
 
 The equation {a) shows that h exceeds unity and increases 
 with positively or negatively increasing values of k. Hence 
 
 The ellipses in which the hyperholoid of one napjje cuts 
 planes perpendicular to its axis of Z are larger the farther the 
 planes are from the centre. 
 
286 OEOMETRY OF THREE DIMENSIONS. 
 
 To find the curves in wliicli the surface intersects planes 
 parallel to the other co-ordinate planes, we transpose either 
 the term in x or that in y, thus putting the equation in the 
 form 
 
 Assigning any constant value to y, we see that the equation is 
 that of an hyperbola, and we may show, as in the case of the 
 other section, that these hyperbolas are all similar. But 
 there is one remarkable case, namely, that in which the equa- 
 tion of the intersecting plane is 
 
 y=±h. 
 
 The equation of the intei'section then becomes 
 
 a c 
 or {ex — az)(cx -j- az) = 0, 
 
 which is the equation of a pair of straight lines. 
 
 This result will be generalized hereafter. 
 
 261. Class III. Hyj^erholoid of Tiuo Nappes. Let two 
 of the terms in (6) be negative. By taking the terms in x 
 and y as negative, and then changing the sign of each mem- 
 ber of the equation, it may be w^ritten in the form 
 
 If c > 2; > — c, the second member will be negative and 
 the equation can be satisfied by no real values of x and y. 
 When z is on either side of the limits ± c, there will be real 
 values of x and y. Hence the surface is composed of two dis- 
 tinct slieets, or naypes, separated at their nearest points hy the 
 sp)ace 2c. This surface is therefore called the hyperboloid 
 of two nappes. 
 
 We readily see that the j^lane z = k, parallel to the plane 
 of XY, intersects the surface in a real ellipse whenever k > c. 
 
 We also show, as in the last section, that the planes x = k 
 and y = h intersect it in hyperbolas. 
 
qUADRIG SURFACES. 287 
 
 Tangent and Polar Lines and Planes to a 
 Qnadric. 
 
 362. Since the equation of the general quadric surface 
 may be reduced to one of the three forms just considered, we 
 may, without loss of generality, consider the equations (6) as 
 representing every such surface. Moreover, we may, in 
 beginning, restrict ourselves to the first form, 
 
 1 + 1+-? = !' (11) 
 
 because the fact that any of the three terms of the first mem- 
 ber has the negative sign may be indicated by substituting 
 
 — a", — J* or — c" for a", ¥ or c^, 
 
 Def. A tangent line to a surface is a line which passes 
 through two coincident points of the surface. 
 
 Pkoblem. To find the condition that a line shall touch a 
 surface of the second order at thepoi?it {x^, y^, z^) on that sur- 
 face. 
 
 Solution. Since the line passes through the point {x^, y^, z^) 
 of tangency, its equations may be written in the form 
 
 X = x^ + Ip; ^ 
 
 y = yi + ^^p; [ («) 
 
 z — z^ -{- nf>, ) 
 
 So long as I, m and n are unrestricted, these equations may 
 represent any line through the point (x^, y^, z^). 
 
 To find the points in which the line meets the surface (11), 
 we must substitute these values of x, y and z in the equation 
 of the surface. Doing this, and arranging the equations in 
 powers of p, we have the condition 
 
 ^r" 4_ y^" , \" 1 4- 2/)f ^^^ 4- ^y^ 4- ^^^^ 
 
288 GEOMETRY OF THREE DIMENSIONS. 
 
 Since the point {x^, ?/„ 2,) lies on the surface by hypothe- 
 sis, we have 
 
 from which it follows that p = is a root of (b). This gives 
 the point {x^, y^, z^ as one of the points, as it ought to. 
 Dividing by p, the equation becomes 
 
 which gives, for the other value of p, 
 
 J Ix. . mil. . nz\ r . vf . w' . ,. 
 
 We have hitherto subjected the line {a) to no restriction 
 except that of passing through the point {x^, y^, z^. The 
 equation (^Z) gives the value of p in terms of I, m and n for 
 the second point in which the line intersects the surface. 
 
 Now, the problem requires that this second point shall 
 coincide with the first one, that is, that p = in {d). This 
 gives 
 
 as the required condition that the line A shall touch the 
 quadric at the point (x^, y^, z^). 
 
 All the quantities except I, m and n in this equation being 
 regarded as given constants, it constitutes a linear equation 
 between I, m and n. Hence, by § 247 {h), it requires that 
 the tangent line lie in the plane 
 
 X 
 
 ^,{x - x^) + -|(y - y.) + ii(2 - z,) = 0, 
 which, by (c), readily reduces to 
 
 Hence we reach the conclusion: 
 
qUADRIC SURFACES. 280 
 
 Theorem XL All straight lines toiccliing a quadric sur- 
 face at the same point lie in a certain plane passing through 
 that jioint. 
 
 Def. The plane containing all lines tangent to a surface 
 at the same point is called a tangent plane to the surface, 
 and is said to touch the surface at that jwint. 
 
 263. Lines upon the Hyperloloid of One Nappe. The 
 result of §260, that a plane may intersect an hyperboloid in a 
 pair of straight lines, is a special case of the following theorem: 
 
 Theorem XII. Through every point upon the hyperloloid 
 of one nappe pass tioo straight lines which lie luholly on the 
 surface, and ivhich form the intersection of the plane tangent 
 at that point tvith the stcrface. 
 
 Proof We may write the equation (9) of the hyperboloid 
 in the form 
 
 (-:+3(-:-a=(^+f)(-i)- (*) 
 
 Now, putting A for an arbitrary constant, let us consider 
 the two planes whose equations are: 
 
 First plane, ^+ i= ^(i + |); ") 
 
 I I 1 , \ (^) 
 
 Second plane, = -Al — ^]. \ 
 
 ^ a c X\ h) } 
 
 Every set of values of x, y and z which satisfies these two 
 equations simultaneously satisfies the equation {h) of the sur- 
 face, as we readily find by multiplication. But all such 
 values belong to the line in which the two planes intersect. 
 Hence this line lies wholly in the surface. 
 
 We have next to show that, by giving a suitable value to 
 X, this line may pass through any point of the surface. Let 
 us put {x^, ^j, ^j), the point through which the line is to pass. 
 
 The factor A must then satisfy the two equations 
 
 5a + .!. = A(l + f); 
 
 a c 
 
 a ~c X\ bl' 
 
290 GEOMETRY OF THREE DIMENSIONS. 
 
 whence 
 
 a c b b a c ^ ^ 
 
 These two equations give the same value of X when a:,, y^ 
 and z^ are required to satisfy the equation of the surface. 
 Substituting the second vahie of X in the first equation (c) of 
 the line of intersection, and the first value in the second, 
 these equations readily reduce to 
 
 Taking half the sum and half the difference of these equations, 
 they become 
 
 (") 
 
 which are still the equations of the line in question, in another 
 form. But the first of these equations is that of the tangent 
 plane. Hence the line lies in the tangent plane as well as on 
 the surface, and therefore forms the intersection of the plane 
 with the surface. 
 
 The other line through {x^, y^, 2,) is found, in the same 
 way, to be given by the simultaneous equations 
 
 ^r^ , y.y 
 
 a' "^ b' 
 
 0' - ^' 
 
 ^1^ , !/ 
 
 ac~^ b 
 
 ac b 
 
 X . z L y\ 
 
 
 
 a c /x\ b J 
 
 
 
 The value of ju, found like that of A, is 
 
 
 
 a c 
 
 a 
 
 z 
 c 
 
 Thus the equations of the second line become the same as 
 
qUADBIC SURFACES. 
 
 21J1 
 
 those already found for the first one, except that the signs of 
 y^ and y are changed. In part, we find 
 
 x,x 
 
 + 
 
 
 
 = 
 
 1; 
 
 
 ac 
 
 — 
 
 y 
 
 x^z 
 ac 
 
 = 
 
 — 
 
 
 (/) 
 
 which are the equations of another line in the surface and 
 passing through (x^, y^), thus proving the theorem. 
 
 264, The equations {e) and (/) represent two lines, 
 each situated both in the surface and in the tangent plane. 
 Hence the theorem may be expressed in the form: 
 
 Theorem XIII. Every tangent plane to the hyperloloid 
 of one nappe intersects the surface in a pair of straight lines 
 passing through the point of tangency. 
 
 It is evident from the preceding theory that the surface in question 
 may be generated by the motion of a line. We present three figures 
 showing the relations which have been discussed. In the first, OP is a 
 central axis or rod, supported on a fixed disk at the bottom and carrying 
 
 a similar disk at the top. The latter can be turned round on the rod. 
 Vertical threads pass from all points of the circumference of one disk to 
 the corresponding parts of the other, thus forming a cylindrical surface. 
 Then turning the upper disk through any angle less than 180°, the 
 threads will form an hyperboloid of revolution, as shown in the other 
 
292 OEOMETRY OF THREE DIMENSIONS. 
 
 figure. The threads shown in the figure are those of one system only ; 
 by rotating the disk in the opposite direction the threads would be those 
 of the other system. 
 
 The next figure represents the surface as cut by a plane very near the 
 tangent plane, the section being an hyperbola of which the transverse 
 
 axis is vertical. By moving the cutting plane a little closer to the 
 centre, the bounding curves of the section will merge into the dotted 
 lines, and the plane will be a tangent to the surface at their point of 
 intersection. 
 
 265. Equations of the Generatitig Lines. To study the 
 lines in question, let us refer each to the point in which it in- 
 tersects the plane of XY. We shall then have z^ = and 
 
 a' ^ ¥ 
 The equations of any one of the first set of lines will then 
 become 
 
 y ^x^ _ III 
 h ac h' 
 
 Because the lines lie in both of the planes represented by these 
 equations taken singly, the coefficients ?, m and 7i in the vec- 
 torial form must, by § 347, satisfy the conditions 
 
QUADBIC SURFACES. 293 
 
 !?'+> = '' 
 
 ac 
 
 These equations give the following values of I and m in 
 terms of oi, which remains arbitrary: 
 
 7 ^^1 
 
 bx, 
 m = — -71. 
 ac 
 
 First system of lines. 
 
 Proceeding in the same way with the second set of lines, 
 we find, starting from the equations (/), 
 
 ^ +%i =0; 
 b ac 
 
 from which 
 
 7 ^Vi 
 
 m = -n. 
 
 ac 
 
 Second system of lines. 
 
 If we give n in both systems the value ale, so as to 
 avoid fractions, the values of the direction-vectors I, m and 
 n will be: 
 
 / I = -a'y,; ( V =+«>,; 
 
 First system: ^ m = + Vx^-, Second system: -j m' = — Z'^o:,; 
 
 ( n = ahc'j ( n' = ahc) 
 
 the values of the second system being distinguished by accents. 
 
 266. Theorem XIV. On an hyperholoid every line of 
 tlie one system intersects all the lines of the other system. But 
 no tiuo lines of the same system intersect each other. 
 
 Proof Retaining {x^, y^, 0) as the fundamental point of 
 any line of the first system, and putting x^ and y^ for the 
 
294 GEOMETRY OF THREE DIMENSIONS. 
 
 values of ;r, and y^ in case of any line of the second system, 
 the condition of intersection of two lines (§ 249) will be 
 
 (mn' — 7n'n){x^ — x^) + {nV - n'l){y^ - yj = 0, (a) 
 
 the third term being omitted because z^ = and z^ = 0. 
 
 If we substitute for m, m', etc., their values, as just given, 
 this equation becomes 
 
 ab'c{x^ + x^)(x^ - X,) + a'bc(y^ + y,){y, - y,) = 0. 
 
 Dividing by a'^'c, we find it reduce to 
 
 a' ^ V \a' ^ b'J 
 
 Now, by hypothesis, {x^, y^ and (x^, y^ are points on the 
 ellipse in which the plane of XY intersects the surface; that 
 is, on the ellipse whose equation is 
 
 cC ^ h' 
 
 Hence the condition reduces to 1 — 1 = 0, which is an iden- 
 tity, showing that the lines intersect. 
 
 Secondly. Let both lines belong to the same system, the 
 one line intersecting the ellipse of the plane of XY in the 
 point {x^, y^), as before, and the other in the point {x^, y^). 
 We shall then have, for the values of the direction-vectors, 
 
 m = + b^x^; m' — + lfx^\ 
 n = dbc\ n' = cibc. 
 
 The condition {a) of intersection then becomes 
 
 Each term being a perfect square is necessarily positive, so 
 that the condition of intersection is impossible. Q. E. D. 
 
 267. Poles and Polar Planes of the Quadric. Let us 
 consider all possible tangent planes which pass through a 
 fixed point {x^, y^, z^) not belonging to the quadric. Let 
 (^15 !/p ^i) ^6 t^6 variable point of tangency on the quadric. 
 
qUADRIC SURFACES. 295 
 
 The equation of the plane tangent at (.Tj, y^, z^) will then be 
 (13). The condition that this plane shall pass through the 
 point (x^, y^, z^) is that the co-ordinates of this point shall 
 satisfy the equation of the plane, which gives 
 
 ^>-o^M. + !4p_i^0. (14) 
 
 a" ' b' 
 
 This is now a condition which the point of taugency (a;,, ?/„ z^) 
 must satisfy as it varies. Being of the first degree, it shows 
 that this point must lie in a certain plane. The equation of 
 this plane may be written 
 
 Def. That plane which contains the points of tangency 
 of all tangents to a quadric which pass through a point is 
 called the polar plane of that point. 
 
 The point is called the pole of the plane. 
 
 Eemark 1. The point of tangency in the aboye case may 
 move along a curve which ^\\\\ then be the intersection of 
 the polar plane and the quadric. 
 
 Eemark 2. The point {x^, y^, z^) may be so situated that 
 no .real tangent plane can pass through it; for example, in 
 the interior of an ellipsoid. The points of tangency (a;,, y^, z^) 
 in (14) will then be entirely imaginary. But the plane (15) 
 will always be real and determinate; only it will not meet the 
 quadric. Hence: 
 
 Theorem XV. To every point in space correspo7ids a 
 defijiite polar plane relative to any quadric. 
 
 Theorem XVI. Conversely, To every plane corresponds 
 a certain pole. 
 
 Proof, Let the plane be 
 
 Ax + By -^ Cz^ D = 0, {a) 
 
 and let a, h and c be, as before, the principal axes of the 
 quadric. Comparing the equation {a) with (15), we see that 
 they become identical if we can have 
 
 ^o__^. .lo__^. 'io__C 
 
 fi^ ~ /)' h''~ />' o' ~ D 
 
296 GEOMETRY OF THREE DIMENSIONS. 
 
 This only requires that we determine x^, y^ and z^ by the 
 conditions 
 
 __«M, __^. -_£!^. 
 
 which always give real values of x^, y^ and z^, and therefore a 
 real pole. Q. E. D. 
 
 Cor. If the plane approach the centre as a limit, D ap- 
 proaches zero as its limit, and x^, y^ and z^ increase indefi- 
 nitely. Hence 
 
 The pole of any diametral plaJie of a quadric is at infinity. 
 
 Notation. If we call any points P, Q, etc., we shall call 
 their polar planes P', Q', etc. 
 
 268. Theorem XVII. If a point lie onaplane, thepole 
 of the plane ivill lie 07i the polar plane of the point. 
 
 Proof Let a point P be {x^, y^, z^) and a point Q be 
 (x^, y^, z^). Then, by (15), the polar plane §' is 
 
 a^ ^ If ^ c- 
 
 Let the point P lie on this plane. Then the co-ordinates 
 x^, y^, z^ must satisfy this equation; that is, 
 
 a' '^ V"'^ & ' 
 
 This equation shows that the co-ordinates (x^^ y^^ z^ satisfy 
 the equation 
 
 which is the equation of the polar plane of {x^, y^, z^). Hence 
 the pole {x^, y^, z^y or Q, lies on this plane. Q. E. D. 
 
 Cor. If any nunriber of points lie in a plane P', iheir 
 polar planes loill all pass through the pole P of that plane. 
 
 Conversely, If any number of planes pass through a point 
 Q, their poles will all lie on the polar plane of Q. 
 
 Theorem XVIII. If any member of planes iiitersect in 
 a straight line, their poles will all lie i7i another straight line. 
 
QUADRIC SURFACES. 297 
 
 Proof. In order that planes may intersect in one line, it 
 is necessary and sufficient that they should all pass through 
 any two points, taken at pleasure, on that line. Let P and 
 Q be two such points. Then — 
 
 Because all the polar planes pass through the point P, 
 their poles all lie somewhere in the polar plane P'; 
 
 Because these planes all pass through the point Q, their 
 poles all lie somewhere in the polar plane Q'. 
 
 Hence these poles all lie on the intersection of P' and Q', 
 which is a straiglit line. Q. E. D. 
 
 Cor. It is readily shown by reversing the course of reason- 
 ing that if any numher of points lie in a straight line, their 
 jjolar planes ivill all pass through another line. 
 
 Def Two lines so related that all poles of planes passing 
 through one lie in the other are called reciprocal polars. 
 
 EXERCISES 
 
 1. If an ellipsoid, an hyperboloid of one nappe and one of 
 two nappes are formed with the same principal axes, a, h, c, 
 it is required to write the equations of their several polar 
 planes relatively to the pole {x^, y^, zj. 
 
 2. In this case show that the polar planes with respect to 
 the ellipsoid and the hyperboloid of one nappe intersect in the 
 plane of XT on the line b'^x^x -\- a^y^y — 1 = 0. 
 
 3. In the same case show that the polar planes with respect 
 to the hyperboloids of one and of two nappes respectively are 
 parallel. 
 
 4. In the same case show that the polar planes with respect 
 to the ellipse and the hyperboloid of two nappes respectively 
 intersect in a line parallel to the plane of XY and intersecting 
 the axis of Z. 
 
 5. Show that if a pole lies on any diameter, the polar plane 
 will be parallel to the diametral plane conjugate to such dia- 
 meter. 
 
 6. Show that if the pole lie on the surface of the quadric, 
 the polar plane will touch the surface at the pole. 
 
298 GEOMETRY OF TUREE DIMENSIONS. 
 
 1. Show that the reciprocal polar of a line tangent to a 
 quadric is another line tangent at the same point, the two 
 tangents lying in a pair of conjugate diametral planes. 
 
 8. If a line is required to lie in a diametral plane, show 
 that its reciprocal polar must be parallel to the diameter con- 
 jugate to that plane. 
 
 Special Cases of Quaclrics. 
 
 269. In all the preceding investigations it has been 
 assumed that the co-ordinates A, B, C of the centre, given by 
 the equations (2), and the quantities I', m', w' and d' in (5), 
 which determine the three principal axes of the quadric, are 
 sM finite and deter mmate. 
 
 Although in the general case this will be true, yet the nine 
 constants which determine the quadric may have such special 
 values that these quantities may be zero or infinite. The 
 complete discussion of these cases would require us to make 
 extensive use of the theory of determinant^, which we wish 
 to avoid; we therefore shall merely point out to the student 
 the possibility of certain special cases. 
 
 370. The Paraboloid. When we solve a system of three 
 equations with three unknown quantities, like (2), each un- 
 known quantity comes out as the quotient of two quantities 
 (compare eq. (3) of § 188 for example), and the denominator 
 of these quotients is the same for all the quantities. If this 
 denominator approaches zero as a limit, the values of A, B 
 and C in (2) will increase without limit. Hence, if this de- 
 nominator vanishes, the centre of the quadric is at infinity. 
 
 In this case the quadric is called a paraboloid. 
 
 271. The Cone. In reducing the original equation to 
 the form (3), the absolute term d' may vanish. In this case 
 the principal axes a, h and c (§ 257) will all vanish (unless 
 some of the quantities V, m' or n' are also zero), and we sliall 
 have the homogeneous equation 
 
 (jx^ -h hf + hz"" + %(}'yz + 2AV'c + 'Ih'xy = 0. (16) 
 
 Def. A cone is a surface generated by the motion of a 
 
QUADIUC SURFACES. 299 
 
 line which passes through a fixed point and continually in- 
 tersects a fixed curve. 
 
 The fixed point is called tlie vertex of the cone. 
 
 The fixed curve is called the directrix of the cone. 
 
 A quadric cone is one whose directrix is a plane locus 
 of the second degree. 
 
 Theorem XIX. Every liomogeneous equation of the 
 second degree has for its locus a quadric cone lohose vertex is 
 at the origin. 
 
 Taking the equation (16), which is a perfectly general one 
 of the kind named in the theorem, we first prove that its locus 
 is a cone having its vertex at the origin, in the following way: 
 
 We take any point at pleasure on the surface (IG); 
 
 "We pass a line through this point and through the origin; 
 
 We then show that this line must lie wholly on the locus. 
 
 Let {x^, y^y z^) be any point on the surface (16). Then 
 every point {Xy y, z) determined by the equations 
 
 X = px^, J 
 
 y = PVv [ ■ . («) 
 
 z = pz^, ) 
 
 will lie on the line passing through the origin and (a;,, y^, z^). 
 Substituting these values in (16) gives, for the condition that 
 the point {x, y, z) shall lie on the locus, 
 
 By hypothesis {x^, y^, z^) satisfies (16). Hence this condition 
 {h) is satisfied for all values of p; hence every point deter- 
 mined by (a) lies on the surface (16); Avhence this surface is 
 so77ie cone. 
 
 Secondly, being of the second degree, (16) represents a 
 quadric surface; whence, by Th. IV., every plane intersects it 
 in a conic, and it is by definition a quadric cone. 
 
 Kemark. For the directrix of the cone we may take its 
 intersection with any plane whatever not passing through the 
 vertex. Let us then take the plane z = c. We then shall 
 have from (16), for the equation of the directrix, 
 
 gx' _|_ ky'' + 2k'xy + 2k'cx + 2g'cy + kc' = 0. (17) 
 
300 GEOMETRY OF THREE DIMENSIONS. 
 
 The coefficients being all independent, this curve may be any 
 conic whatever. Hence (16) may represent any quadric cone 
 whose vertex is at the origin. 
 
 2*72. Special Case lolien a Quadric becomes a Pair of 
 Planes. Since the directrix of the cone may be any conic, it 
 may be a pair of straight lines. Since a line turning on a 
 point and intersecting a fixed line describes a plane, it fol- 
 lows that lohenever the directrix is a pair of lines, the quadric 
 cone becomes a pair of planes. Hence among the special kinds 
 of quadrics must be included a pair of planes. 
 
 The quadric equation of a given pair oi planes is readily 
 found. If the equations of the planes are 
 
 ax -\- by -{- cz -\- d =0, 
 a'x + b'y + c'z -f d' = 0, 
 
 we liave only to take the product of these equations, which 
 will be of the second degree in x, y and z. 
 
 273. Surfaces of Revolution. In the reduction of the 
 general quadric, two of the principal axes, a and J for example, 
 may be found equal. In this case the equation may be re- 
 duced to one of the forms 
 
 a c 
 
 or 
 
 a c' 
 
 x' + f±a'[^,±lj==0. (17) 
 
 Assigning any constant value to z, the equation in x and y 
 will be that of a circle. Hence all planes parallel to the plane 
 of XFwill intersect the quadric in circles having their centres 
 on the axis of Z. Since all sections containing the axis of Z 
 will be conies, the surface can be generated by the revolution 
 of some conic around the axis of Z. It is therefore called a 
 surface of revolution. 
 
 The equation (17) admits of the same four-fold classifica- 
 tion as the equation (6), according to the algebraic signs of 
 the ambiguous terms. We have therefore, as the three real 
 forms — 
 
QUADRIG SURFACES. 301 
 
 I. The ellipsoid of revohdioji : 
 
 II. Tlie hyperboloid of revolution of one nappe: 
 
 III. The hyperboloid of revolution of two nappes: 
 
 When, in the ellipsoid, 
 
 c > a, the ellipsoid is called prolate ; 
 c < «, the ellipsoid is called oblate ; 
 c — a, the ellipsoid is called a sphere. 
 
 In the hyperboloid of one nappe the axis c may be infi- 
 nite. The equation will then be 
 
 x^ ^ if ^ a\ 
 
 the equation of a cylinder of radius «, whose axis is that ofZ. 
 
 274. Deriving Surf aces from the Generating Curve. The 
 general method by which we find the equation of the surface 
 generated by revolving a curve around the axis of Z is this: 
 
 Assume the curve to be in any initial position. 
 
 Take any point upon it whose vertical ordinate is z, and 
 find the corresponding values of x and y, and hence of 
 Vx" + ?/", in terms of z, the distance of the point from the 
 plane of XY. 
 
 Since this distance remains constant while the point re- 
 volves, the square of the equation thus found will be the 
 equation of the surface. 
 
 If the fixed position can be so chosen that the generating 
 curve may lie wholly in the plane of XZ (or YZ), one of the 
 co-ordinates y or z will then be zero, and we have only to sub- 
 stitute Vx" + if for X or y, as the case may be. 
 
302 GEOMETRY OF THREE DIMENSIONS. 
 
 275, The Paraholoid of Revolution. Let us suppose a 
 parabola to revolve about its principal axis, wliicli we shall 
 take as the axis of Z. The square of each ordinate will then 
 be 2pz, But this square, as the curve revolves, is continually 
 equal to x" -\- y', because the ordinate is a line perpendicular 
 to the axis of Z, whose terminus on the curve is represented 
 by the co-ordinates x and y of the curve. Hence the equa- 
 tion of the surface is 
 
 x' -\-y- = 2pz', 
 2) being the semi-parameter of the generating parabola. 
 
 EXERCISES. 
 
 1. Find the equation of the cone generated by revolving 
 around the axis of Z the straight line whose equation, when 
 tRe line is in the plane XZ, is 
 
 X = mz -f i- 
 Find also the vertex of the cone. 
 
 Aiis. x^ -\-^f=: m'z' + 2mbz + b\ 
 
 Vertex at the point, (o, 0, ). 
 
 2. Investigate the surface generated by the motion of a 
 straight line around an axis which does not intersect it, the 
 shortest distance of the line from the axis being a, and the 
 angle between them being a. (In the initial position we may 
 suppose the line to intersect the axis of JT at right angles at 
 the distance a from the origin, and to form an angle a, whose 
 tangent is m, with the axis of Z.) Find the equation of the 
 surface and its principal axes. 
 
 A71S. x^ -j- y^— m'^z' = a^. Axes: a, a, —. 
 
 Here, if we take a point on the line at the distance r from the axis 
 of X, its co-ordinates in the initial position will be 
 
 x = a; 2 = 7*coscr; y = ?'sin a = 2tan a = ws. 
 
 3. Find the equation of the cone generated by the revolu- 
 tion of the straight line whose equations in one position are 
 
 x = mp; y = 0] z = np. 
 
 Ans. n\x' + y') - m'z' = 0. 
 
qUADRIC SURFACES. 303 
 
 4. Find the equation of the ellipsoid generated by the re- 
 Tolution of the ellipse b'^x^ -\- a^z^ = a^b^, 
 
 6. If the hyperbola cV — a^z' = aV revolve about the 
 axis of Z, find the equation of the curve and of the cone de- 
 scribed by the asymptotes. 
 
 a c 
 
 a' e ~ 
 
 6. Investigate the surface when the revolving hyperbola 
 is a^z" — c'x^ — a^y^. 
 
 7. Find the equation of the surface generated by the revo- 
 lution about the axis of Z of the line whose equations are 
 
 X = a -\- lft\ 
 y z=l) J^mp', 
 z =■ c -\- np. 
 
 Ans. n\x^ + y') = {na - Uy + {nh - mcY + (P + m')z' 
 
 -\- 2(nla -\- mnh — Pc —■ m^c)z. 
 
 8. Show that the equation of a sphere of radius r whose 
 centre is at the point (a, h, c) is 
 
 (X - ay + {^J- by -\-{z- cy - r' = 0, 
 
 and find the value of r in order that the origin may bisect the 
 radius passing through it. 
 
 9. Find the plane of the circle in which the spheres 
 
 (x-ay-^iy-by-\-{z-cy =r' 
 and (x - ay+ {y - by^ {z - cy = r" 
 
 intersect each other. 
 
 10. Show that if three spheres mutually intersect each 
 other, the planes of their three circles of intersection pass 
 through a line perpendicular to the plane containing the 
 centres. (One of the centres may be taken as the origin. ) 
 
 11. Investigate the locus of the equation 
 
 a and b being both positive. 
 
304 GEOMETRY OF THREE DIMENSIONS. 
 
 12. Do the same thing for the equation 
 
 a b 
 
 13. Show that the six planes in which four circles taken 
 two and two intersect each other all pass through a point. 
 
 14. Investigate the relations of the three surfaces 
 
 a' ^ b' 
 
 -5 = '. 
 
 
 -^:=« 
 
 
 -^-'' 
 
 and 
 
 Show that if these surfaces be cut by a plane parallel to 
 that of XY, the two areas included between the three ellipses 
 of intersection will each be constant and equal to the area of 
 the ellipse in which the first surface intersects the plane of 
 XF. 
 
 15. A straight line moves so that three fixed points upon it 
 constantly lie in the three co-ordinate planes. Find the locus 
 of a fourth point upon it whose distances from the other three 
 points are a, b and c. 
 
 16. From the results of §266 deduce the following con- 
 clusions: 
 
 I. The cosine of the angle between the two generating 
 lines through the point (.r^, y^, 0) of the surface is 
 
 a^b\' - a'y^ - b\^ 
 a'b'c' + ay; + ^W 
 
 II. At the ends of the respective axes a and b the cosines 
 are -^—. — tv and 
 
 c' + l^ ""^ c» + a'' 
 III. If a = 6 = c, the lines are at right angles to each 
 other. 
 
PART III. 
 
 INTRODUCTION TO MODERN 
 GEOMETRY, 
 
 276, The Principle of Duality. In modern geometry 
 every straight line is supposed to extend out indefinitely in 
 both directions, and is called a line simi)ly. Hence lines, like 
 points, differ only in situation. 
 
 Def. A segment is that portion of a line contained be- 
 tween two fixed points. Hence a segment is what is called a 
 finite straight line in elementary geometry. 
 
 There are certain propositions relating to lines and points 
 which remain true when we interchange the words jyoiVi^ and 
 line, provided that we suitably interpret the connecting words. 
 
 The principle in virtue of which this is true is called the 
 principle of duality. 
 
 Two propositions which differ only in that the words point 
 and line are interchanged are said to be correlative to each 
 other. 
 
 The following are examples of correlative propositions and 
 definitions. The right-hand column contains in each case 
 the correlative of the proposition found at its left. 
 
 I. Prop. Through any I. P7'op. On any line 
 point may pass an indefinite may lie an indefinite number 
 number of liries. oi points. 
 
 II. Def. Any number of II. Def. Any number of 
 lines passing through a point points lying on a line is called 
 is called a pencil of lines, a row of points, or simply 
 or simply a pencil. a point-row or row. 
 
306 
 
 MODERN GEOMETRY. 
 
 The common point of a 
 pencil is called the vertex 
 of the pencil. 
 
 III. Prop. Two points 
 determine the position of a 
 certain line, namely, the line 
 joining them. 
 
 IV. Def. The line join- 
 ing two points is called the 
 junction-line of the points. 
 
 V. Prop. Three points, 
 taken two and two, determine 
 by their junction-lines three 
 lines. 
 
 VI. Prop. A collection 
 of n lines, taken two and two, 
 
 has, m general, -^^ — 
 
 junction-points. 
 
 VII. Def. A collection of 
 four lines, with their six junc- 
 tion-points, is called a com- 
 plete quadrilateral. 
 
 The line on which a row 
 of points lie is called the car- 
 rier of the row. 
 
 III. Prop. Two lines de- 
 termine the position of a cer- 
 tain point, namely, their point 
 of intersection. 
 
 IV. Def. The point of in- 
 tersection of two lines is called 
 the junction-point of the 
 lines. 
 
 V. Prop. Three lines, 
 taken two and two, determine 
 by their intersections three 
 junction-points. 
 
 VI. Prop. A collection 
 of n points, taken two and 
 
 two, has, in general, -^^-^r — - 
 
 K) 
 
 junction-lines. 
 
 VII. Def. A collection 
 of four points, with their six 
 junction-lines, is called a 
 complete quadrangle. 
 
 ^^""Z 
 
 9 
 
 y^ 
 
 2 
 4 
 
 VIII. Prop. On each of 
 the four sides of a complete 
 quadrilateral lie three ver- 
 tices. 
 
 VIII. Prop. Through 
 each of the four vertices of a 
 complete quadi'angle 
 three sides. 
 
PRINCIPLE OF DUALITY. 
 
 307 
 
 IX. Prop. The complete 
 quadrilateral has three diago- 
 nals, formed by joining the 
 junction-point of each two 
 sides to the junction-point of 
 the remaining two sides. 
 
 If two lines are represented 
 by the symbols a and h, their 
 junction-point is represented 
 by ah. 
 
 The pencil of lines from a 
 vertex P to the points a, h, c, 
 etc., is represented by P-ahc, 
 etc. 
 
 When two lines are each 
 represented by a pair of point- 
 symbols, a comma may be in- 
 serted between the pairs when 
 their junction-point is ex- 
 pressed. 
 
 Ex. The expression ah, xy 
 means the junction-point of 
 the lines ah and xy. 
 
 IX. Prop. The complete 
 quadrangle has three minor 
 vertices, being the intersection 
 of the junction-line of each 
 two vertices with the junction- 
 line of the remaining two. 
 
 If two points are repre- 
 sented by the symbols a and 
 h, their junction-line is repre- 
 sented by ah. 
 
 The row of points in which 
 a carrier R intersects the lines 
 A, B, G, etc., is represented 
 by R-ABC, etc. 
 
 When two points are each 
 represented by a pair of line- 
 symbols, a comma may be in- 
 serted between the pairs when 
 their junction-line is ex- 
 pressed. 
 
 Ex. The expression AB, 
 XY means the junction-line 
 of the points AB and XY. 
 
 Scholium. When, in elementary geometry, two intersecting lines 
 are drawn, their junction-point, being evident to the eye, is not sepa- 
 rately marked. But when two points are given, it is considered neces- 
 sary to draw their junction-line wherever this line is referred to. But 
 this is not always necessary in the higher geometry, and such lines 
 may be omitted when drawing them would make the figure too compli- 
 cated. 
 
308 MODERN GEOMETRY. 
 
 The Distance-Ratio and its Correlative. 
 
 377. The Distance- Ratio. Heretofore the position of a 
 point on a straight line has been expressed by its distance 
 (positive or negative) from some other point, supposed fixed 
 on the line. 
 
 The position of the point may also be expressed by the 
 'ratio of its distances from two fixed points on the line. 
 
 Let a and h be two fixed points on an indefinite line, 
 which points we may regard as the ends of a segment ai of 
 the line. Let x^, x^ and x^ be three positions of a movable 
 point X, and let us consider the ratio ax : hx of the distances 
 of X from the points a and b. If we put 
 
 h, the distance ab; 
 h, the distance ax\ 
 r, the ratio ax : bx. 
 
 we shall have 
 
 K 
 
 (1) 
 
 This fraction, or ax : bx, is called the distance -ratio 
 of the point x "with respect to the points a and h. 
 
 NoTATioiq^. The distance-ratio is written 
 
 . _ax 
 (a.b,x)^-^-. 
 
 Let us now study the changes of value of the distance- 
 ratio as the point moves along the line. 
 
 Assuming the positive direction to be toward the right, 
 then, when x is in the position x^, the distances ax^ and bx^ 
 will both be positive, and we shall have 
 
 r>+l.f ^' 
 
THE DISTANCE-RATIO. 305^ 
 
 If X recedes indefinifcely toward the right, k increases in- 
 definitely and the ratio t — y appronclies unity as its limit. 
 
 Tlierefore, for a point at infinity on the line, we have 
 
 r = + 1. 
 Supposing the point to move toward the left, the denomi- 
 nator k — li will become zero when x reaches h; and as tliis 
 point is approached, the fraction r will increase without limit. 
 Hence, when x is at Z>, 
 
 r = 00. 
 
 When X is in the position x^ between a and h, ax will be 
 positive and Ix negative. Hence, in this part of the line, 
 r = a negative quantity. 
 As X passes from h to a, r will increase from negative in- 
 finity to zero. 
 At a, 
 
 r = 0. 
 
 In the position x^ both terms of the ratio will be negative, 
 and ?• will be a positive proper fraction. 
 
 As X recedes to infinity on the left, r will approach unity 
 as its limit. Hence, whether we suppose x to reach infinity 
 ill the negative or positive direction, we have, at infinity, 
 
 r = + 1, 
 
 and no distinction is necessary between the two infinities. 
 
 If, then, we suppose the point .t to move along the whole 
 line from negative to positive infinity, we may consider it as 
 arriving back at its starting-point, and being ready to repeat 
 the motion. During this motion the distance-ratio r will 
 also have gone through all possible values from negative to 
 positive infinity, and will be back at its starting-point. The 
 order of positions of the jooint and the order of changes of r 
 are as follows: 
 
 Point: Infinity; negative; at point a\ on fine a&; at point h; posi- 
 tive; infinity. 
 Dist. r.: Unity; positive < 1; zero; negative; infinity; positive > 1; 
 unity. 
 
310 
 
 MODERN GEOMETRY. 
 
 278. To exhibit to the eye the changes in r as a; moves 
 along the hne, we may erect at each point of the line an ordi- 
 nate the length of which shall represent the value of r at 
 that point. The curve passing through the ends of the ordi- 
 nates will be that required. 
 
 The ratio r being a pure number, the length which shall 
 represent unity may be taken at pleasure. So we may lay 
 down from the middle point m of ab an arbitrary length 
 772^ = — 1, and the lengths of all the other ordinates will be 
 fixed. 
 
 2*79. Theorem. T7ie position of a point is completely 
 fixed hy its distance-ratio with r'espect to tzvo given points; 
 that is, there can he only one point on the line to correspond to 
 a given value of the distance-ratio. 
 
 This is the same as saying that, in the equation (1), only 
 one value of k will correspond to given values of r and h. 
 This is readily proved by solving the equation with respect to 
 h. We note that in the special case when r = 1 the point is 
 at infinity. 
 
 280. Relation of the Distance- Ratio to the Division of a 
 Line. The conception of the distance-ratio occurs in elemen- 
 tary geometry when we say that the point x divides the line 
 al internally or externally into the segments ax and hx, having 
 
THE SINE-RATIO. 311 
 
 a certain ratio to each other. This ratio is identical with the 
 distance-ratio just defined. It is negative when the line ab is 
 cut internally; positive when it is cut externally. 
 
 We may therefore, instead of saying, '^ The distance-ratio 
 of the point x with respect to the points a and h," say, 
 '* The ratio in which the point x divides the segment ah.*^ 
 
 EXERCISES. 
 
 1. Show that the curve which expresses the value of r in 
 the preceding section is an hyperbola; find its asymptotes; 
 define the class to which it belongs; construct its major axis 
 in the case when we take mq = ab. 
 
 2. Show that if we take two points at equal distances on 
 each side of the middle point m of the base-line, the pro- 
 duct of the corresponding values of r will be unity. Trans- 
 late this result into a property of the equilateral hyperbola. 
 
 281. The Sine- Eatio, In the two preceding articles we 
 showed how to express the position of a varying jt?o^n^ upon a 
 fixed line. The correlative of this problem is that of express- 
 ing the position of a varying line which must pass through a 
 fixed poi7it. 
 
 As, in the first case, the position of the moving point is 
 expressed by its relation to two fixed points on the line, so, 
 in the second case, the position of the moving line is fixed by 
 its relation to two fixed lines passing through the point. Let 
 us put ^, X 
 
 0, the fixed point; i 5^ 
 
 OA, OB, the two fixed | / / \ 
 
 lines; Bl.^..,^^^ 1 / 'O^-^^ 
 
 OX, the moving line. X^^^^^^^I^^^^j^-^^^^^' 
 
 From any point P of this j^^-^"^'^/^^^ P^^^-^~-_ 
 line drop the perpendiculars / 
 
 FF' and FF'' upon the fixed / 
 
 lines. Let us then consider the ratio 
 
 FP' 
 
 pp,r 
 
312 MODERN GEOMETRY. 
 
 We readily see that the value of r is the same in whatever 
 position on the line OX the point P is chosen, and that 
 
 sin A OX 
 
 R = 
 
 sill BOX' 
 
 Hence we call E the sine-ratio of the line OX tvith re- 
 spect to the lines OA and OB. 
 
 To investigate the algebraic signs of sin A OXand sin ^ OX, 
 let us take the directions OA, OB and OP as positive. Then, 
 in accordance with the usual trigonometric convention, the 
 sine of A OP will be positive or negative according as a person 
 standing at and facing toward A has the point P on the 
 left or right side of the line OA. 
 
 Suppose the line OX to start from the position OA and to 
 turn round in the positive direction. Then, 
 
 As OX starts from OA, 
 
 R starts positively from zero. 
 When OX reaches the bisector OX', 
 
 because then AOX -{- BOX = 180°. 
 As OX approaches the position OB', 
 
 R increases indefinitely, 
 
 because sin ^ OX approaches zero. 
 When OX reaches OB', 
 
 R = oo. 
 As OX passes from OB to the bisector OX", 
 
 R increases from — oo to — 1. 
 When OX reaches OX", 
 
 R= -1. 
 As OX passes from OX" to OA', 
 
 R increases from — 1 to 0. 
 
 The line OX has now reached its initial position, though 
 its positive direction is reversed. Completing the revolution. 
 
DISTANCE- AND SINE-RATIO. 313 
 
 we see tluit R goes through the same scries of vahies as before. 
 Hence 
 
 The sine-ratio depends 07ily upon the 2)Osition of the mov- 
 ing line, and is the same whethej' we take one direction or the 
 other as positive. 
 
 282. Division of the Angle, As, in § 280, we have sup- 
 posed the point x to divide the line ah, so we may in the jire- 
 ccding construction suppose the line OX to divide the angle 
 BOA into the parts BOX ^i\^ AOX. We then take for the 
 dividing ratio, not the ratio of the angles themselves, but that 
 of their sines. 
 
 Note. The student may remark a certain incongruity when we 
 speak of the point x dividing the Hne ab into the segments ax and bx, 
 because it is not the algebraic sum but the algebraic difference of the 
 8eii;ments which makes up the Hne ab. This incongruity would be 
 avoided by measuring one of the segments in the opposite direction, 
 making x its initial point, thus taking ax and xb as the segments. But 
 it is more convenient to take x as the terminal point of each segment, 
 and to accept the incongruity of calling a line the algebraic difference 
 of its parts, because no confusion will arise when the case is once under- 
 stood. 
 
 The same remarks apply to the division of the angle, 
 
 283. Distinction of Aiitecedent and Consequent in Dis- 
 tance- and Sine-Ratio. In forming a ratio one of the terms 
 must be taken as the antecedent (or dividend), and the other 
 as the consequent (or divisor). By interchanging the points a 
 and b the antecedents and consequents will be interchanged, 
 and the ratio will therefore be changed to its reciprocal. 
 
 To give clearness to the subject we shall employ the fol- 
 lowing notation: 
 
 The points a and h from which we measure the segments 
 ax and Ix will be called hase-points. 
 
 That base-point from which the antecedent segment of the 
 ratio is measured will be called the A-point. 
 
 That base-point from which the consequent segment of the 
 
 ratio is measured will be called the B -point. 
 
 ax 
 Then, when the ratio ~ is represented in the form 
 
 {a, I, X), 
 
314 MODERN GEOMETRY. 
 
 we write first the A-point, next the B-point, and lastly the 
 terminal point, which we may call the T-point. 
 
 284. Permutation of Points. If we use the notation 
 
 jp = length ab, 
 q El length hx, 
 
 we shall have, by the definition of the distance-ratio. 
 
 Let us represent this ratio by the symbol r. Then, by 
 permuting the base-points between themselves (that is, by 
 making a the B- and h the A-point), we shall have 
 
 {h, a, x) = — ^ = i, (J) 
 
 a result which we may express by the general proposition: 
 
 I. By permuting the base-points we change the distance- 
 ratio into its reciprocal. 
 
 By permuting b and x in («), we have 
 
 (a,x,b) = f^ = -^ = l-r. (c) 
 
 That is: 
 
 II. By permuting the B- and T-points we change the dis- 
 tance-ratio r into 1 — r. 
 
 The same permutation applied to {b) gives 
 
 (b, x, a) = — = — T— n^ — r = — n — = . (d) 
 
 ^ ' ' ' xa — {p -i- q) P -\- q r 
 
 Lastly, by permuting the base-points in (c) and (c?), 
 
 {x, a, b) = ^-—^; (e) 
 
 (a:, b, a) = — ^. (/) 
 
DISTANCE- AND SINE-RATIO. 
 
 315 
 
 EXERCISES. 
 
 1. By comparing the forms {a) and (d), show that if, in 
 the expression 
 
 cpr 
 
 r - 1 
 
 we put q)r for r and repeat the substitution in the result, we 
 shall get r itself. 
 
 2. Find the distance from the A-point (§ 277) of points 
 whose distance-ratios are 
 
 2'^2'^2* 
 
 A 2 
 Ans. -p) 
 
 -V\ ^P- 
 
 3 1 
 
 If «, h, X and y be any points whatever, show that 
 (a, hy x) _ (i, a, y) _ (x, y, a) _ {y, x, h) 
 
 4. 
 of 
 
 (a, h, y) {h, a, x) {x, y, h) 
 Show that if from the vertex 
 an isosceles triangle abc we 
 draw a line ex to the base, the sine- 
 ratio in which the angle c is divided 
 by the line ex equals the distance- 
 ratio in which the base ab is cut by 
 the point x. 
 
 In algebraic language the theorem is 
 
 a)- 
 
 a^ 
 
 sm acx 
 
 ax 
 hx' 
 
 sin bcx 
 
 5. If the angle A OB is 120°, in what directions must 
 those lines be drawn which will divide the angle in the respec- 
 tive sine-ratios — 2 and + 2? 
 
 6. If the point x divide the segment ah in the ratio -j- 1 : 2, 
 in what ratio will h divide the segment ax^ Ans. 2 : 3. 
 
 7. If the points x and y divide the segment ai in the re- 
 spective ratios + 2 and — 2, in what ratios will a and b re- 
 spectively divide the segment xy? Ans. + 3 and — 3. 
 
 8. If the sum of the distance-ratios of two points, x and y, 
 is unity, show that ax x ay — ah"*. 
 
316 
 
 MODERN GEOMETRY. 
 
 Theorems involving the Distance- and Sine- 
 
 Katios. 
 
 385. Def, If each of the sides or angles of a polygon is 
 divided by a point or line, the ratios of the divisions are said 
 to be taken in order when each vertex is a divisor-point for 
 one of its sides and a dividend-point for the other side. 
 
 If the divisions are all internal, we shall, in going round 
 the polygon, have the divisor- and dividend-segments in alter- 
 nation. 
 
 386. Theorem I. If any 
 three lines le drawn from the 
 three vertices of a triangle to its 
 opposite sides, the contiiiued 
 product of the sine-ratios in 
 which the angles are divided isc' 
 equal to the continued product 
 of the distance-ratios i7i which 
 the sides are divided, the ratios 
 leing all talcen iyi order. 
 
 Hypothesis. A triangle ahc of which the sides and angles 
 are divided by the lines ax, ly and cz. 
 
 Conclusion. If we put 
 
 r^, r^, 7*3, the distance-ratios in which the sides are divided 
 by the points x, y and z respectively; 
 
 R^, R^, R^, the sine-ratios in which the angles are divided 
 by the respective lines ax, ly and cz, we have 
 
 ^1 ^\ ^ = ^1 ^. ^3- 
 
 Proof By the theorem of sines in trigonometry we have, 
 in the triangles hax and cax, 
 
 ex _ sin cax ^ 
 ax ~ sin c ' 
 hx _ sin hax 
 ax ~ sin h 
 
DISTANCE- AND SINE-RATIO. 
 
 317 
 
 Dividing the first equation by the second, 
 ex sin cax sin b 
 
 In the same way we find 
 
 hx~ 
 
 sin bax 
 
 sin c 
 
 find 
 
 
 
 ^Z- 
 
 sin aby 
 
 sin c 
 
 cy 
 
 sill cby 
 
 sin«' 
 
 bz 
 
 sin bcz 
 sm «6';Z 
 
 sin a 
 
 az 
 
 sin b' 
 
 Taking the continued product of the three last equations, we 
 have 
 
 ex ay bz _ sin eax sin aby sin bcz 
 
 bx 
 
 ay 
 ~cy 
 
 az 
 
 sin bax sin eby sm acz 
 
 The three fractions in the first member of this equation 
 are the distance-ratios in which the sides are divided, and 
 those in the second member are the sine-ratios in which the 
 angles are divided, so that the theorem is proved. 
 
 28 1*. Theorem II. The continued product of the dis- 
 tance-ratios in which any transversal cuts the sides of a tri- 
 angle is equal to unity. 
 
 Proof Let a transversal 
 cut the sides of the triangle 
 abc in the points x, y and z. 
 
 Through any vertex, as^ 
 b, draw a line parallel to the 
 transversal, meeting the op- 
 posite side in the point b'. 
 
 Then, forming the distance-ratios in which the sides bo 
 and ba are divided, using the similar triangles thus con- 
 structed, we have 
 
 az _ ay 
 Tz~t~y' 
 bx _ b'y ^ 
 
 ex 
 
 while 
 
 cy 
 
 cy ^ c/y_ 
 ay ay 
 
318 MODERN GEOMETRY. 
 
 The continued product of these equations gives 
 
 ££.*?.. oy. ^ 1. Q. E. D. 
 Dz ex ay 
 
 Remaek. Since the demonstration takes no account of 
 algebraic signs, we have not yet shown whether the product 
 is + 1 or — 1. It is evident that the transversal must cut 
 either two sides of the triangle internally, or none. Hence 
 either two factors or none at all will be negative; whence the 
 product is always positive and equal to + 1. 
 
 Corollary. If three points in a straight line he tahen on 
 the three sides of a triangle, the junction-lines from each point 
 to the opposite ve7iex divide the a7igles into parts the continued 
 product of luhose sine-ratios is imity. 
 
 For, by Th. I., the product of the sine-ratios is equal to 
 that of the distance-ratios, and, by Th. 11. , the continued 
 product of the latter is unity. 
 
 288, Theorem III. Conversely, If on the three sides 
 of a triangle abc lue take any three j^oints x, ?/, z, such that 
 
 az hx cy _ 
 
 bz ' ex ' ay ' 
 
 these points will he in a straight line. 
 
 Proof Let z' be the point in which the line xy cuts the 
 side ah of the triangle. We shall then have, by Th. I., 
 
 az' hx cy _ 
 hz' ' ex' ay ~ 
 
 Comparing with the equation of the above hypothesis, we 
 find 
 
 az _ az\ 
 
 Tz~ U'' 
 
 that is, the distance-ratios of the points z and z' with respect 
 to a and h are the same. 
 
 Because there is only one point on ab which has a given 
 distance-ratio, the points z and z' are coincident and z hes 
 on the line xy. Q. E. D. 
 
DISTANCE- AND SINE-RATIO. 
 
 319 
 
 289, Theorem IV. If three lines 2^assi7ig through a 
 point be draiun from the vertices of a tri 
 angle, the angles will be so divided that the 
 sine- ratios, taken in order, will be — 1. 
 
 Proof. If ABC be the triangle, and 
 P the point, we have, in the triangles 
 PAB, PBO and PCA, neglecting alge- 
 braic signs, 
 
 sin BAP _AP^ 
 sin ABP ~ BP' 
 s in CBP _BP^ 
 sin BCP~ OP' 
 sin ACP _ CP 
 sin GAP ~ AP' 
 
 The continued product of these equations would give 
 sin^^P sin CBP sin ACP 
 
 sin CAP ' sin ABP ' sin BCP~ 
 
 ±1. 
 
 (a) 
 
 Algebraic signs having been neglected, it remains to be 
 found whether this product is positive or negative. We have 
 the theorems: 
 
 I. Lines drawn from any point within a triangle to the 
 three vertices cut the angles internally. 
 
 II. Of the three lines drawn to the vertices of a triangle 
 from an external point, and produced if necessary, two will 
 divide the angles internally and one externally. 
 
 I is evident. To 
 prove II let the whole 
 plane without the tri- 
 angle be divided by its 
 sides into the six regions 
 A,A',B,B', CandC". 
 Then the angle whose 
 sides bound A will be 
 cut internally or exter- 
 nally, according as the 
 point is situated within or without one of the regions A and 
 
320 MODERN GEOMETRY. 
 
 A\ Considering the otlier angles in the same way, we see 
 that onl}^ one angle can be cut internally and that the other 
 two will be cut externally. 
 
 The sine-ratio being positive for an external and negative 
 for an internal division, either one or all three of the factors 
 in {a) must be negative. Hence 
 
 sin BAP sin CBP sui A CP _ 
 sin CAP ' sin ABP ' sin BCP ~ * Q. E. D. 
 Corollary. Three lines passing from the vertices of a tri- 
 angle through a 2^oint cut the opposite sides so tliat the con- 
 tinued product of the distance-ratios, taken in order, is nega- 
 tive unity. 
 
 For, by Theorem I., this product is equal to that of the 
 corresponding sine-ratios, which product is negative unity, by 
 the theorem. 
 
 390. Theorem V. Conversely, If three points cut the 
 respective sides of a triangle so that the continued product of 
 the distance-ratios is negative unity, the lines joining these 
 2Joi7its to the opposite vertices of the triangle yass through a 
 point. 
 
 Proof If aic be the triangle, and x, y and z be the points, 
 we have, by hypothesis, 
 
 az hx cy _ 
 bz ' ex ' ay 
 Join the points x and y to the opposite vertices, a and b, 
 of the triangle by lines intersecting at a point 0. From c 
 draw a line L through 0, and let z' be the point in which it 
 cuts be. 
 
 By Th. IV., Cor., we then have 
 az' bx cy _ 
 bz' ' ex ' ay 
 Comparing with the hypothesis, we have 
 az _ az' 
 Tz ~ W 
 Therefore the points z and z' are coincident and the line 
 from z to c is identical with L, and so passes through the 
 point in which ax and by intersect. Q. E. D. 
 
Q 
 
 ANHARMONIC RATIO. 321 
 
 EXERCISES. 
 
 1. Explain wliiit Tlicorem 11. shows when the transversal 
 is parallel to one of the sides. 
 
 2. What does I'heorcni IV. become when the point 
 through which the lines are drawn is at infinity? 
 
 3. Show by Theorem V. that the three medial lines of a 
 triangle pass through a point. 
 
 4. Show by the preceding theorems what bisectors of the 
 interior and exterior angles of a triangle meet in a point. 
 
 5. If from the vertices at the ends 
 of the base BC ot a triangle we draw 
 lines intersecting on the medial line 
 AQ and meeting the opposite sides in 
 the points B^ and C, show that B'C 
 is parallel to BC. 
 
 6. In this case what relation exists B' 
 between the distance-ratios in which the sides AB and^C 
 are divided by the points B' and C? 
 
 The Anliarmonic Ratio. 
 
 391. Taking any point x on a J t y 
 
 the line ab, we have, by what precedes, a distance-ratio ax : Ix 
 or (rt, h, X) of the point x with respect to the jioints a and h. 
 In the same way, taking a fourth point y, w^e have a distance- 
 ratio («, J, y). Then: 
 
 id 1) xS 
 
 Def. The quotient ; ' / — f of the distance-ratios of the 
 («, ^, y) 
 points X and y with respect to the points a and h is called the 
 anhariuonic ratio of the four points a, J, x and y. 
 
 Tliat terminal point x which enters into the numerator of 
 the fraction wdll be called the A-T-point; the other, the B-T- 
 point. 
 
 It will be seen that the anharmonic ratio is a pure number 
 ■whose value depends upon the mutual distances of the four 
 points. 
 
 292. The following are simple corollaries from the de- 
 finition of the anharmonic ratio: 
 
822 MODERN GEOMETRY. 
 
 I. If the terminal points are ioth outsiae the segment ah, 
 or both within it, the anharmojiic ratio is positive. 
 
 For in the first case the distance-ratios are both positive, 
 and in the second they are both negative. 
 
 II. If one terminal point is witJmi and the other without 
 the segment ah, the anharmonic ratio is negative. 
 
 For the two distance-ratios then have opposite signs. 
 
 III. If the two terminal points coincide, the anharmonic 
 ratio is unity. 
 
 IV. If three points, namely, the hase-points and one ter- 
 minal point, are fixed, while the other terminal point may move, 
 then for every value which we may assign to the anharmonic 
 ratio there will ie one and only one position of the movable 
 point. 
 
 For if we put r — the anharmonic ratio, and suppose the 
 points a, b and x to be fixed, we have, by definition. 
 
 whence 
 
 {a, b, x) ' 
 (a, h, y) = (a, b, x) X r. 
 
 Now, the points a, b and x being fixed, the quantity 
 {a, b, x) is a constant, so that for every different value we 
 assign to r we shall have a different value of the distance- 
 ratio {a, b, y), and hence a different position of the point y 
 (§279). 
 
 It will be seen that the four points which enter into an 
 anharmonic ratio form two pairs, one pair being the two base- 
 points, the other pair the two terminal points. The two 
 points of each pair are said to be conjugate to each other. 
 
 ia b x'\ 
 Notation". We represent the anharmonic ratio ) ' / — { 
 
 («, ^, y) 
 
 in the form 
 
 {a, b, X, y). 
 
 Expressing the points by their general designations 
 (§§ 283, 291), the order of writing them is 
 
 (A, B,A-T,B-T). 
 
ANHARMONIC RATIO. 
 
 Writing the ratios at length, we have 
 
 , , . ax '. hx ax .by . . 
 
 ia, by X, y) = -~ = —-. (a) 
 
 ^ ' ' ' ^' ay: by ay . bx ^ ' 
 
 293. Permutation of Points. Let us now consider the 
 problem, What changes will result in the anharnionic ratio 
 by interchanging the different points? 
 
 By interchanging the two base-points, that is, by making 
 b the A-point and a the B-point, we change each distance- 
 ratio into its reciprocal (§ 284, I), and hence the anharmonic 
 ratio into its reciprocal, because we always have 
 
 l:q p' 
 
 whatever be p and q. 
 
 The same result follows by interchanging the terminal 
 points, because we then change the terms of the fraction 
 (a^^x) .^^^ {a, b, y) 
 (a, b, y) {a, b, x)' 
 
 Hence, if we make both changes, the anharmonic ratio 
 will be restored to its original value. 
 
 If we simply make the base-points the terminal ones, and 
 vice versa, the anharmonic ratio is unaltered. For, by the 
 notation, 
 
 xa : ya _ xa : xb 
 
 {x, y, a, b) 
 
 xb : yb ya : yb^ 
 
 which is identical with (a), the signs of each of the four seg- 
 ments being changed. 
 
 It follows from this that there will be four permutations 
 which will leave the anharmonic ratio unchanged, and four 
 others which only change it to its reciprocal. They are as 
 follows: 
 
 {a, b, X, y) = {b, a, y, x) = {x, y, a, b) = {y, x, b, a) = r; (1) 
 (5, a, X, y) = {a, b, y, x) = {x, y, b, a) = (y, x, a, b) = -.(2) 
 
324 MODERN GEOMETRY. 
 
 In all these permutations the four points are paired in the 
 same way, a and b being one pair and x and y the other. 
 
 Hence the eight permutations which do not change the 
 pairing of the conjugate-points can only interchange the terms 
 of the anharmonic ratio. * 
 
 When the pairing of the points is changed, a may ha-ve 
 either x or y as its conjugate-point. To find the effect of 
 these permutations, we start from the following identical 
 equation which always subsists between the six segments ter- 
 minated by the four points «, h, x and y. These segments 
 are al), ax, ay, bx, by and xy. h h + j^ 
 
 ax . by + <^^ • y^ + <^y • ^^ = ^' (^) 
 
 To proYG this equation, Ave substitute for ab and ay their 
 values 
 
 ab = ax + ^bf 
 
 ay = ax -\- xy, 
 
 and so write the first member of the equation in the form 
 
 ax.by -\- ax yx -\- ax xl), 
 
 -\- xb + ^y 
 which is the same as 
 
 ax{by -i- yx -\- xb) + xb{yx + xy), 
 an expression which vanishes identically, because 
 
 by -\- yx + ^^ = 0; yx -{- xy E 0. 
 
 Now divide (a) by ay .bx. We thus find 
 
 ax . by ab . yx _ ^ 
 ay.bx ay .bx ~ ' 
 that is, 
 
 (a, b, X, y) -f {a, x, b, y) = I. (b) 
 
 Hence, using the same notation as before, 
 
 (a, X, b, y) =1- r. 
 
 * In tills pairing process note the analogy of conjugate-points to 
 partners at -whist. There are eight arrangements of the players around 
 the table which will not change the pairing, and there are three ways 
 in which the players may choose partners. 
 
ANHARMONIC RATIO. 325 
 
 We now liave, in the same way as before, 
 
 («, X, h, y) = {x, a, y, b) = (b, y, a, x) = (y, b, x, a) = 1 - r; (3) 
 
 {x, a, b, y) = (a, x, y, b) = {y, b, a, x) = {b,y, x, a) = ^--— -. (4) 
 
 We have finally to consider the case in which a is paired 
 with y. To pair a with y, we remark that the equation [b), 
 being true whatever points we suppose a, b, x and y to rep- 
 resent, may be considered a brief expression of the theorem: 
 
 By inter clianging the B- and A-T-i)oints, we form a new 
 anharmonic ratio tuliich, added to the original one, makes 
 unity. 
 
 Applying this theorem to the second expression in line (4), 
 
 it gives 
 
 («., X, y, b) + {a, y, x, b) = 1. 
 
 1 r 
 
 Hence {a, y,x,b) — 1 — {a, x, y, b) = 1 — = 
 
 r r — 1' 
 and 
 
 {a, y, X, b) = {y, a, b, x) = {b, x, y, a) = {x, b, a, y) = ^r^; (5) 
 
 r — 1 
 (y, a, X, b) = (a, y, b, x) = {x, b, y, a) = {b, x, a, y) = —^. (6) 
 
 The equations (1) to (6) include all 24 permutations of 
 a, b, X and y, which, however, give rise to only G different 
 values of the anharmonic ratio, namely, 
 
 1 _ 1 r r — 1 , . 
 
 r \ — r r — 1 r 
 
 294. The preceding operations lead to a curious algebraic 
 result. Suppose that, instead of starting with the equation 
 (1), we had started with any of the others, (G) for example. 
 We could then have obtained expressions for the remaining 
 20 anharmonic ratios by performing the same operations on 
 (6) which we have actually performed upon (1), only the ex- 
 pression would have taken the place of r all the way 
 
 through. But, if the process is correct, we should then arrive at 
 the same expressions for the other 20 anharmonic ratios which 
 
326 MODERN GEOMETRY. 
 
 weliave actually found. The same being true if we start from 
 any other of the six equations, we conclude: 
 
 If, in the set of expressions (c), we substitute for r any one 
 expression of the set, the values of the several expressions zvill 
 he cha7iged into each other in such a way that the set will re- 
 main unaltered except in its arrangement. 
 
 As an illustration, let us substitute the sixth expression, 
 
 r — \ 
 
 , for r all the way through the set (c). Then, by reduc- 
 tion of the fractions, 
 
 r — 1 
 
 r will be changed to 
 
 r 
 
 1 r 
 
 — will be changed to 
 
 r r — 1' 
 
 1 — r will be changed to — ; 
 
 will be changed to r; 
 
 - will be changed to 1 — r; 
 
 will be changed to . 
 
 r 
 r — 
 r-1 
 
 We have thus reproduced the same set, only differently 
 arranged. 
 
 295. Anharmonic Ratio of a Pencil of Lines. As we 
 have formed the anharmonic ratio of four 
 points on a line by their distance-ratios, so 
 we may form the anharmonic ratio of 
 four lines passing through a point by 
 means of their sine-ratios. 
 
 The four lines A, B, X and Y pass . i 
 through the point P. If we take A and \^ 
 
 B as the base-lines forming the angle APB, the line X will 
 give the sine-ratio 
 
 BmAPX 
 liiiBPX' 
 
Aiikar^(pulc ^aY/os^ 
 
 
 
 
 V^ 
 
 P^'P^' P^<^ Si^x 0( ^JUy<5' 
 
 AP' ^ 8V^- ^^c^(,^^ ^ 
 OA 03 OP oP ^ 
 
 z. 
 
 K 
 
 JX Xt^O<> 
 
 Ax _ AI^ . BP 
 
 7<^ 
 

ANHARMONIC RATIO. 
 
 327 
 
 and the line Fwill give the sine-ratio 
 
 si n APY 
 sin BPY' 
 
 The quotient of these ratios, or 
 
 sin APX . sin APY _ sin .^PX sin BP Y 
 sin BPX ' sin BPY ~ sin BPX sin APY' 
 
 is called the anharmouic ratio of the pencil of lines PA, 
 PB,PXaiidPY. 
 
 Designating each line by a single letter, we may write 
 
 (A, B, X, Y) 
 
 as the anharmonic ratio of the four lines A, B, Xand Y. 
 
 296. FuKDAMENTAL THEOREM. If a transversol cvoss 
 a pencil of four lines, the anharmonic ratio of the four points 
 of intersection icill he equal to the anhar^nonic ratio of the 
 pencil. 
 
 Proof Let ABXYhQ the 
 pencil, intersecting the trans- 
 versal ay in the points a, l, x 
 and y. 
 
 We begin, as in § 286, by 
 comparing the distance-ratio" 
 {a, h,x) with the sine-ratio {A, B, X). 
 
 From the equations 
 
 ax : Px = sin aPx : sin xaP^ 
 bx : Px = sin hPx : sin xbP, 
 
 we obtain, by division, 
 
 ax _ sin aPx sin xbP^ 
 bx ~ sin bPx ' sin xaP' 
 
 or, using the abbreviated notation, 
 
 sin xbP 
 
 {a, b, x) = (A, B, X) 
 
 sin xaP' 
 
328 MODERN GEOMETBT. 
 
 We find in the same way 
 
 Taking the quotient of these equations, 
 
 (a, h a) _ (^, B, X ) 
 K b, y) (A, B, YY 
 
 The first member of this equation is, by definition, the 
 anharmonic ratio of the four points a, h, x and ?/, while the 
 second is that of the pencil of lines A, B, JTand Y. Thence 
 
 (a, h, X, y) = {A, B, X, Y). 
 
 Q. E. D. 
 
 Cor. 1. If any number of transversals cross the same 
 pencil of four lines, the anharmonic ratios of the four ^^oints 
 of intersection on the several transversals will all be equal. 
 
 For each such ratio will be equal to the anharmonic ratio 
 of the pencil. 
 
 Cor. 2. If from a roiv of four points lines be drawn to a 
 fifth movable point, the anharmonic ratio of the pencil thus 
 formed ivill be C07istant, lohatever be the position of the fifth 
 point. 
 
 For the anharmonic ratio of the pencil will be constantly 
 equal to that of the row. 
 
 Scholium. Using the notation of § 276, IX., the preced- 
 ing propositions may be expressed as follows: 
 
 If P be any vertex, and a, b, x, y a row of points, then 
 
 Anh. ratio (P-a,b,x,y) = (a, b, x, y). 
 
 If p be any transversal, and A, B, X, Y the four lines of 
 a pei^cil, then 
 
 Anh. ratio {p-A,B,X,Y) = {A, B, X, Y). 
 
 297. Application of the Principle of Duality. The 
 branch of Plane Geometry which we are now treating is 
 subject to the principle of duality (§ 276); that is, 
 
 From every proposition respecting the relations of points 
 and lines we may form a second correlative proposition respect- 
 
ANUARMONIG PROPERTIES. 329 
 
 ing the relation of lines and points, by interchanging the words 
 line and point, as follows: 
 
 Line instead of Point. 
 
 Junction-point of ) ^^ (Junction-line of two 
 
 two lines ) \ points. 
 
 Point on a line '^ Line through a point. 
 
 Pencil of lines " Row of points. 
 
 Three points in a ) ^ j Three lines through a 
 
 line ) ( point. 
 
 Anharmonic ration ( Anharmonic ratio of 
 
 of hues throudi V '' ■{ - , ,. 
 
 . , ° I ] points on a line. 
 
 a point ; ( 
 
 The correlative proposition is not necessarily different 
 from the original one. When the two are identical, the pro- 
 position is self-correlative. 
 
 The relation of the proposition to its correlative is mutual; 
 that is, the correlative of the correlative is the original pro- 
 position. 
 
 To make the notation correlative we represent the junction- 
 point of the two lines A and B by AB. 
 
 Let us, as an example, change the preceding fundamental 
 proposition into its correlative. The two then read: 
 
 ^. ( pencil of four lines t r.<.,i (line. 
 
 Given ; a -^ ^ „ ^ . and any fifth ■< , ' 
 
 ( row 01 four points ( point, 
 
 TVe conclude : the anharmonic ratio of the four 
 
 jnnction- \ ^"'^ of the four \ ^'^.^^ with tlie fifth \ ''".<^ 
 ( lines ( points ( point 
 
 equal to that of the given \ P®^^^ • 
 
 ( row. 
 
 By reading the top lines Ave have the original proposition; 
 l)y reading the bottom lines, its correlative. By making the 
 construction it will be seen that the correlative proposition is 
 identical with the original one. 
 
 The principle of duality applies to the demonstration as 
 well as to the proposition. By making the above substitu- 
 tions the demonstration of the original becomes the demon- 
 
 is 
 
330 
 
 MODERN GEOMETRY. 
 
 stration of the correlative. It is tlierefore in rigor not 
 necessary to give the latter; and when we do so, it is only to 
 assist the student. 
 
 398. Theorem. If tve have two lines intersecting m a 
 point p, and if we have on the one line any three points 
 a, h, c, and on the other line three points a', b', c' , such that 
 the anharmonic ratio (p, a, b, c) is equal to (p, a', b', c'), 
 then the three junction-lines aa', bb' and cc' meet in a point. 
 
 Proof. Let if and iVbe the given lines, and let q be the 
 junction-point of the lines aa' and bb\ Join qp and qc', and 
 let c" be the junction-point of the line iV^ with the line qc\ 
 
 Then, because M and N are two transversals crossing the 
 pencil q-a',p,h',c', we have 
 
 [p, a, b, c") = {p, a\ b\ c'). 
 
 By hypothesis, 
 
 (p, a, b, c) = (p, «', b% c'). 
 Hence 
 
 (p, a, b, c") = {p, a, b, c). 
 
 The points j9, a and b being given, there is only one fourth 
 point which can form with them a given anharmonic ratio 
 (§ 292, IV.). Hence the points c and c" are coincident, 
 and the junction-line c'c is identical with c'c", and so passes 
 through the point q, in which, by construction, aa' and bb' 
 intersect. Q. E. D. 
 
ANHARMONIC PROrEBTIES. 331 
 
 Correlative Theorem. If we have two pomfs on a line 
 Q, and if tlivougli one point pass three lines A, B and 
 C, and through the other point pass three lines A\ B* and C", 
 such that the anharmonic ratio (Q, A, B, C) equals 
 {Q, A', B'f C), then the three junction-points A A', BB' and 
 CC lie in a straight line. 
 
 Proof Let m and n be the points; let P be the junction- 
 line of the points A A' and BB'y c the junction-point PC\ 
 and C" the junction-line nc. 
 
 Then, because the pencils Q, A*, B\ C and Q, A, B, C" 
 pass through the same four points of the line P, we have 
 
 {Q,A,B, 0") = (Q,A',B',C"). 
 
 But, by hypothesis, 
 
 (Q,A',£',C") = (Q,A,B,C). 
 TTen oe 
 
 (Q,A,B,Cn = {Q,A,B, C) 
 
 These equal anharmonic ratios having three lines identical, 
 the fourth lines C and (7" are also identical; whence the lines 
 C and C" intersect at c, the junction-point C^P; whence the 
 junction-points AA\ BB' and CC" all lie on the line P, 
 
 Q. E. D. 
 
 Note. The student should compare the demonstration, step by 
 step, with that of the original proposition, and note the relation of each 
 step. 
 
332 MODERN GEOMETRY. 
 
 Projective Properties of Figures. 
 
 299. Let there be a j^oint 0, a plane P and a figiire Q 
 each situated in any position in space. If lines (called lines 
 of projection) pass from to each point of Q, the points in 
 which these lines intersect the plane P form a second figure 
 which is called the projection of the figure Q. 
 
 This definition of a projection is more general than that of elemen- 
 tary geometry, in which the lines of projection are all parallel to each 
 other and perpendicular to the plane P. The latter is a special case in 
 which the point is at infinity in a direction perpendicular to the plane. 
 
 It may be remarked that the shadow of a figure upon a plane, as 
 cast by a luminous point, is identical with its projection. But should 
 the distance of any part of the figure from the plane exceed the distance 
 of the luminous point, there could be no shadow, but there would still 
 be a projection, formed by continuing the lines of the rays in the 
 reverse direction, namely, from the figure through the luminous point. 
 
 300. The following are some simple relations between 
 figures and their projections: 
 
 I. The projection of a point is a point. 
 
 II. The projection of a straight line is a straight line. 
 For since the straight line and point lie in a plane, the 
 
 lines of projection are all in this plane, and the projection is 
 the intersection of this plane with the plane of projection. 
 
 III. The projection of a roiu ofpoi^its is another row ivhose 
 carrier is the projection of the original carrier, 
 
 IV. The projection of a pencil is a 2^e7icil. 
 
 V. The projection of a curve and a tangent is another 
 curve and a tangent. 
 
 VI. Every 2^rojection of a line passes through the point in 
 which the line intersects the plane of projection. 
 
 VII. The projection of a circle is a conic section. 
 
 For the lines from a point to the circumference of a circle 
 form the elements of a cone. Hence their intersection with 
 the plane of projection is the intersection of a conical surface 
 with that plane, and is therefore, by definition, a conic section. 
 
 VIII. If the projected figure Q is in a plane P', and if we 
 call Q' its projection on the plane P, then Q itself is the pro- 
 jection of Q' upon the plane P'. 
 
ANHARMONIG PROPERTIES. 
 
 333 
 
 This follows at once from the definition, the lines of pro- 
 jection being identical in the two cases. 
 
 IX. Every section of a circular cone can he projected into 
 a circle. 
 
 For, by taking the vertex of the cone as tlie point 0, and 
 its circular base as the plane of projection, the outline of this 
 base becomes the projection of any section of the cone. 
 
 301. Theorem. The projection of a roiu of foicr ptoints 
 has the same anhar7nonic ratio as the original row. 
 
 Proof, The lines of projection 
 of the four points form, by defini- 
 tion, a pencil having its vertex at 
 0. The carriers, both of the origi- 
 nal and the projected row, form 
 
 transversals crossing this pencil, a' V c' a' 
 
 and the two rows of points are the intersections of these 
 carriers with the lines of the pencil. The anharmonic 
 ratios of the two rows are therefore equal (§ 296, Cor. 1). 
 
 Q. E. D. 
 
 302. Theorem. The projection of a pencil of four lines 
 has the same anharmo7iic ratio as the original pencil. 
 
 Proof Let 0-ahcd be the ^0 
 
 given pencil, and let a, h, c and d 
 be the points in which it intersects 
 the plane of projection. 
 
 Because the lines of the pencil 
 must all lie in one plane, the 
 points a, h, c and d will lie in a 
 straight line. 
 
 If 0' be the projection of 0, the projected pencil will be 
 O'-ahcd. 
 
 Then the anharmonic ratios of each pencil will be equal 
 to («, 1), c, d), and so will be equal to each other. 
 
 Remark. Those properties and relations of a figure 
 which remain unchanged by projection are called projective 
 properties and relations. 
 
334 MODERN GEOMETRY. 
 
 Harmonic Points and Pencils. 
 
 303. Def. When the anharmonic ratio of four points is 
 negative unity, tliey are called a row of four harmonic 
 points, and each pair of conjugate points is said to divide 
 the segment joining the other pair harmonically. 
 
 So a pencil of four lines of which the anharmonic ratio is 
 negative unity is called an harmonic pencil. 
 
 Cor. The anharmonic ratio being negative, one of the 
 terminal points must divide the base-line internally and the 
 other must divide it externally. Hence the order of the four 
 points is such that the conjugate points of the one pair, a and 
 l, alternate with those of the other pair, x and y. 
 
 If the point x is half way between a and h, its conjugate, 
 y, is at infinity. 
 
 If X then move toward h, y will also move toward h from 
 the right, and the two points will reach h together. 
 
 If X move toward a, y will approach from infinity on the 
 left, and the two points will reach a together. 
 
 The law of change is expressed by the following theorem: 
 
 304. Theokem. The product of the distances of two ter- 
 minal harmonic points from the middle of the base-line is con- 
 stant, and eqtial to the square of half the base-line. 
 
 Proof. The condition that the anharmonic ratio of the 
 four points a, b, x, y = — 1\b 
 
 ax ^ ay _ 
 
 bx ' by ~ ' 
 
 which is equivalent to 
 
 ax.by -\- ay . bx = 0. 
 
 Let m be the middle point of ab. Then 
 
 ax = am + mx; 
 
 by =■ bm-\- my = — am -\- my, 
 
 ay = am -\- my; 
 
 bx = — am -\- nix. 
 
\ 
 
 HARMONIC POINTS AND LINES. 335 
 
 By substitution tlie equation (a) reduces to 
 
 — 2{a?ny + ^mx.my = 0. 
 Hence 
 
 mx.my=(am)\ (1) Q. E. D. 
 
 Eemark. On the line ab we may take any number of 
 
 ' ' — I , I „ i .„ ' — nr, rr, 1' 
 
 X X z y y y 
 
 pairs of points, x and y, fulfilling the condition (1), and 
 therefore dividing harmonically the segment ab. 
 
 Def. Three pairs of points which divide harmonically the 
 same segment are said to form an involution. 
 
 305, The Fourth Harmonic. 
 
 Def. When three points of an harmonic row are given, the 
 fourth is called the fourth harmonic of the other three. 
 
 Problem. Having given three poi7its of an harmonic roiv, 
 to find the fourth. 
 
 Construction. ' Let a, h and c be 
 the given points, and let a and h be 
 the conjugate base-points. 
 
 On the middle point rn of ai 
 erect a perpendicular mp = \ah, and '^ 
 on the other side of m from c take mc' = mc. 
 
 Through j9 and c' describe a circle having its centre upon 
 the line ab. 
 
 The other point, d, in which this circle cuts ab will be the 
 fourth harmonic required. 
 
 Proof. From eq. (1) and from the theorem of elementary 
 geometry which gives c'm.md =^ {mpY the proof is readily 
 found. 
 
 306. Fourth Harmonic of a Pencil. When three lines 
 of a pencil are given, the fourth line necessary to form an 
 harmonic pencil may be constructed. 
 
 From § 296 it follows that every pencil of four lines passing 
 through a row of four harmonic points is an harmonic pencil, 
 and, conversely, that every transversal intersects an harmonic 
 pencil in four harmonic points. 
 
336 MODERN GEOMETRY. 
 
 Hence, to construct the fourth harmonic of a given pencil 
 of three lines, we draw any transversal, find the fourth 
 harmonic of the three points of intersection, and join it to 
 the vertex. 
 
 30')'. Harmonic Properties of the Triangle, 
 Theorem. If from a point are drawn three lines to the 
 vertices of a triaiigle, and at any two of the vertices the fourth 
 harmonics to the lines thence emanating are constructed, these 
 fourth harmonics luill rneet C/ 
 
 the line from the third ver- 
 tex in a point. I \ \^ '"*""'';^p' 
 
 Proof Let ABC be 
 the triangle, and P the 
 point. Let CP' and BP' 
 
 be the fourth harmonics of A^^^^ ^^^=^'B 
 
 the pencils C-APB and B-APG. 
 
 Then, because AP, BP and CP meet in a point, 
 
 sin BAP sin CBP sin ACP _ _ .. . 
 
 sm CAP ' sin ABP ' sin BCP ' ^^ ' ^ 
 
 Because BP^ and CP' are fourth harmonics conjugate to 
 BP and CP respectively, 
 
 sin CBP _ _ sin CBP \ 
 sin ABP ~ sin ABP'' 
 sm ACP smACP' 
 
 sin BUP sin BCP'' 
 
 By substitution in the equation {a) we have 
 sm BAP sin CBP' sin ACP' 
 
 = - 1. 
 
 sin CAP ' sin ABP' ' sin BCP' 
 
 Therefore the three lines AP, BP' and CP' meet in a 
 point (§§286, 290). Q. E. D. 
 
 Scholium. By drawing the fourth harmonic at A and 
 considering the other two pairs of vertices, A, B and C, A, we 
 have two other points of meeting, making four in all. 
 
 If the point P is the centre of the inscribed circle, the 
 lines from P will be the bisectors of the angles, and the 
 three points P' will be the centres of the escribed circles. 
 
HARMONIC POINTS AND LINES. 
 
 337 
 
 308. Correlative Theorem. If three points on a line 
 he taken 07i the sides of a triangle, and the fourth harmonics 
 to two of them be constructed, these fourth harmonics loill he 
 on a line luith the third point. 
 
 Proof. Let ahc be the triangle; 
 
 X, y and z, three points on the sides in a right line; 
 
 x', y' and z' , the fourth harmonics to the rows b, c, x; 
 c, a, y; a, h, z, respectively. 
 
 Because x', y' and z' are fourth harmonics, 
 
 7 "^ 
 
 az' 
 
 hx^^ 
 
 cy' 
 
 iP) 
 
 az _ 
 
 hz~ 
 
 hx _ 
 
 ex ~ 
 
 cy _ 
 
 ay ~ ay 
 Because x, y and z are on a right line, 
 az hx cy _ 
 bz ' ex ' ay~ 
 
 By substituting for any two of these factors their values 
 from {b), we prove the theorem, by § 288. 
 
 EXERCISES. 
 
 1. Any two orthogonal circles cut the line joining their 
 centres in four harmonic points. 
 
 2. Any circle of a family having a common radial axis 
 cuts harmonically the common chord of the orthogonal 
 family. 
 
338 MODERN OEOMETRT. 
 
 Aiiliarmonic Properties of Conies. 
 
 309. Lemma. If two tangents to a circle are intersected 
 hij a third tangent, the points of intersection subtend from 
 the centre of the circle an angle riieasured by one half the arc 
 letioeen the tioo tangents. p \y 
 
 Proof. Let be the centre 
 of the circle; 
 
 m, n, the points of tangency 
 of the two tangents; | 
 
 p, the third point of tan- 
 gency; 
 
 m', ?i', the points of intersec- . . 
 
 tion. / 
 
 It is then easily shown, from the equality of the lines m'p 
 and m'm, that Om' is perpendicular to pni. 
 
 In the same way, 0?i^ is perpendicular to p7i; 
 
 Angle m' On' = angle mp7i. 
 
 By a fundamental property of the circle. 
 
 Angle mp}i = -J angle 7uOn. 
 Therefore 
 
 Angle m' On' = i angle ?nOn. 
 
 Q. E. D. 
 Cor. If the third tangent pn' moYes around the circle, 
 the angle m' On' will remain constant, being always equal to 
 \m On. 
 
 310. Theorem. If fonr fixed tangents touch a conic, 
 and a inovable fifth tangent intersect them, the anliarmonic 
 ratio of the four points of intersect ioji is the same for all posi- 
 tions of the fifth tangent. 
 
 Proof. Let the conic be projected into the circle whose 
 centre is 0, and let a, h, c and d be four points in which 
 the fifth tangent intersects four fixed projected tangents, 
 touching the circle at the points t^, t^, t^, t^. 
 Because Angle aOb = i arc-angle tf^, j 
 
 Angle bOc = i arc-angle tj, V (g 309) 
 
 and Angle cOd — ^ arc-angle tj^, ) 
 
ANUABMONIG PROPERTIES OF CONICS. 
 
 339 
 
 and t^, /„ etc., are, by hypothesis, fixed, these angles remain 
 constant however the fifth tangent may move. 
 
 Therefore the anharmonio ratio of the pencil 0-abcd re- 
 mains constant, being a function of the sines of constant 
 angles. 
 
 Therefore the anharmonic ratio of the row a, h, c, d re- 
 mains constant (§296). 
 
 Because this anharmonic ratio is constantly equal to that 
 of the corresponding points in the projected figure (§ 301), 
 the latter also remains constant. Q. E. D. 
 
 311, Theorem. If from foiir fixed points of a come 
 lines le draimi to a fifth variable point, the anharmonic ratio 
 of the pencil thus formed will remain constant whatever the 
 position of the fifth point. 
 
 Proof. Project the conic into a cir- 
 cle. Let a, hy c and d be the projec- 
 tions of the four fixed points, and P 
 that of the fifth point. 
 
 By a fundamental property of the 
 circle the angles aPh, hPc, etc., will re- 
 main constant as P moves on the circle. 
 
 Therefore the anharmonic ratio of 
 the pencil P-abcd will remain constant. 
 
 Therefore the anharmonic ratio of the corresponding pen- 
 cil in the conic will remain constant (§ 302). Q. E. D. 
 
 Dff. The constant anharmonic ratio of a pencil whose 
 
340 MODERN OEOMETRT. 
 
 lines pass from four fixed points on a conic to any fifth point 
 is called the anharmoiiic ratio of the four points of the conic. 
 Def. The constant anharmonic ratio of the points in which 
 four fixed tangents to a conic intersect a fifth tangent is called 
 the anliarmo7iic ratio of the four tangents to the co7iic. 
 
 312. Extension of the Principle of Duality to Ctirves. 
 If we conceive a series of points to follow each other according 
 to some law, their junction-points will form a broken line or 
 a polygon. If each point of the series approaches indefinitely 
 near to the preceding one, the broken line approaches a curve 
 as its limit. We may therefore define a curve as the limit of 
 a series of junction-lines when the points approach each other 
 indefinitely. 
 
 The correlative conception, 
 on the principle of duality, 
 will be that of a series of lines 
 following each other according 
 to some law, and approaching 
 each other indefinitely. The 
 junction-points of consecutive^ 
 lines will be the correlatives of 
 the broken lines, and as they approach each other indefinitely 
 they will tend to lie on some curve as their limit. 
 
 In the first case, if we suppose the points to be consecu- 
 tive positions of a moving point, this point will move on the 
 limiting curve. 
 
 In the correlative case, if we suppose the lines to be the 
 consecutive positions of a moving line, this line will con- 
 stantly be tangent to the limiting curve. 
 
 We thus have, as correlative conceptions: 
 
 Points on a curve corresponding to Tangents to a curve. 
 
 Junction-point of ) ^ .. j Junction-line of two 
 
 two tangents ) ( points, i.e., a chord. 
 
 Points in which a) ( rn . ^ 
 
 ... . , / ,, \ Tangents from a 
 
 line intersects a }- " < - , ^ .1 
 
 \ ) point to the curve, 
 
 curve ; I 
 
rASCAL'S THEOREM. 341 
 
 313. Pascal's Theorem. If a hexagon he inscribed in 
 a conic, the three junction-points of its three pairs of opposite 
 sides lie in a straight line. 
 
 Eemark. By a polygon inscribed in a curve is meant any 
 chain of straight lines, returning into itself, whose consecutive 
 junction-points all lie on the curve. A polygoii of n sides 
 may be formed by taking any n points and joining them con- 
 secutively in any order whatever. 
 
 a X a' a' 
 
 Proof. Let 1 2 3 4 5 6 be the inscribed hexagon, of which 
 the opposite sides are 
 
 12 and 45; 
 23 and 5 6; 
 3 4 and 6 1. 
 
 Select three alternate vertices, as 2, 4 and 6, and consider 
 the pencils 
 
 2-1345 and 6-134 5. 
 
 Because these pencils are formed by joining the four fixed 
 points 1, 3, 4, 5 on the conic to the points 2 and 6 respective- 
 ly, their anharmonic ratios are equal (§ 311). 
 
 Let a, h, c, d be the row of points in which the pencil 
 2-13 45 intersects the line 45. We shall then have 
 
 {a, h, c, d) = Anh. ratio (2-1345). (§ 296) 
 
 Lei; a', ¥, c, d' be the row of points in which the pencil 
 6-1345 intersects the line 3 4. We have 
 
 (a', l\ c, d') = Anh. ratio (6-13 45). 
 Hence 
 
 (a, hf c, d) = {a', h', c, d'). 
 
342 MODERN GEOMETRY. 
 
 These two rows have the point c common to both, and this 
 point occupies the same position in the two ratios. There- 
 fore the three lines 
 
 aa', hi', dd' 
 meet in a point (§ 298). 
 
 But a and a' are the respective junction-points (1 2 and 4 5) 
 and (6 1 and 3 4), while hb' = ^d and dd' = 5 6. 
 
 Hence the opposite sides 23 and 5 6 intersect on the line 
 aa' which joins the junction-points of the two other pairs of 
 opposite sides. Q. E. D. 
 
 314. Correlative of Pascal's Theorem: Brian- 
 chon's Theorem. The three lines joining the opposite ver- 
 tices of a hexagon cirmimscrihed about a conic meet in a point. 
 
 Proof. Let the sides taken in order be 1, 2, 3, 4, 5, 6. 
 
 Consider the two rows of points a, Z>, c, d and a', b', c' , d' 
 in which the sides 1, 3, 4 and 5 intersect 2 and 6. 
 
 Because the tangents 2 and 6 are each intersected by the 
 four tangents 1, 3, 4, 5, we have 
 
 (a, b,c,d) = (a', b', c\ d'). (§ 310) 
 
 Consider the pencil A-a,b,c,d. We have 
 
 Anh. ratio {A-a,b,c,d) = (a, b, c, d). (§ 296) 
 

 
 J. i _ j_ 
 
 
 u 
 
 K'p 'P,'P' 
 
 _ ^^ 
 
 
 Z' 
 
"1 = 1^^ S^ 0^ 
 
 X^-/^ ^ ^ -Lax- '^^y. -f- ^%^ i>'^'-r '^^ o 
 
 =z O 
 
 
 y^ 
 
 
 i 
 
TRILINEAR CO-ORDINATES. 343 
 
 Also, for the same reason 
 
 Anh. ratio {A' -a' ,b\c' ^cV) = (a', b\ c', cV). 
 
 Hence 
 
 Anil, ratio (A-a,b,c,d) = Anli. ratio (-!'«', ^', c', cV). 
 
 Now these two pencils have the line Ac = A'c' common, 
 and occupying the same position in the two ratios. Hence 
 the three remaining lines intersect in three points in a right 
 line (§298, Cor.); that is, the three points 
 
 (e) where diagonal Act crosses diagonal A' a'-, 
 
 (b) where line Ab crosses A'y; 
 
 {d') where line Ad crosses A'd\ 
 are in a right line, which proves the theorem. 
 
 Trilinear Co-ordinates. 
 
 315. In the method of trilinear co-ordinates the posi- 
 tion of a point is defined by its relation to the three sides of 
 a general triangle. Distances from each side are considered 
 positive when measured in the direction of the opposite ver- 
 tex; negative in the opposite case. 
 
 Theorem. If a general triangle be given, the itosition of 
 a point is completely determined tvhen the mutual ratios of 
 its distances from the three sides of the triangle are given. 
 
 Proof. Let it be given that ^fsA. 
 
 the distances of a point P from 
 
 the sides AB and ^C of a tri- cX / V 
 
 angle are in the ratio m : n. 
 
 If we draw a line Ax divid- 
 ing the angle BA C in the sine- B^^^ — -^ ^ C 
 
 ratio , every point of this line will fulfil the given con- 
 dition (§281). 
 
 If it be also given that the distances of the point P from 
 the sides ^Cand BA are in the ratio p ; m, then the point 
 P must also lie on the line By dividing the angle B in the 
 
 sine-ratio — — . 
 m 
 
344 MODERN GEOMETRY. 
 
 Hence, if both ratios be given, the only point which will 
 satisfy them is the junction-point of the lines Ax and By, 
 which is therefore the required point P. Q. E. D. 
 
 Method of Expressing the Ratios of Distances. The 
 mutual ratios of the distances of a point P from the three 
 sides of a triangle are most conveniently expressed by three 
 numbers proportional to these distances. Let us i3ut 
 
 6/j, d^, 6Z3, the distances of P from the three sides of the 
 triangle; 
 
 x^, .^2, x^, any three numbers proportional to d^, d^, d^. 
 We then have 
 
 (1) 
 
 
 x^ : X, : x^^d,\ d. 
 
 '•d,, 
 
 d: 
 
 _ 5.. d, _x,^ 
 x^ ' d, x^ ' 
 
 d,_ _ d,_ _ d. 
 
 d^ X, 
 
 also, -'- = -"- = -^. (3) 
 
 ^1 ^2 ^3 
 
 If we put p for the common value of the three fractions 
 (3), we have 
 
 d^ = fix^\ J 
 
 d^ ^ px^'\ (4) 
 
 The sets of equations (1), (2), (3) and (4) are so many 
 different ways of expressing the fundamental fact of the pro- 
 portionality of the numbers x^, x^ and x^ to the distances d^, 
 d^ "and dy 
 
 316, Relation hetween the Distances. The position of 
 the point is completely determined when its actual distances 
 from any two sides of the triangle are given. Hence from two 
 distances the third may be found, which shows that there is 
 some equation between the distances. If we put 
 
 a, h, c, the lengths of the three sides of the fundamental 
 triangle; 
 
 A, the area of the triangle, 
 the equation in question is 
 
 ad^ + hd^ -f cd^ = 2A. (5) 
 
TRILINEAR CO-ORDINATES. 845 
 
 This equation is readily proved by drawing lines from the 
 point to the three vertices and equating the algebraic sum of 
 the areas of the three triangles thus formed to that of tlie 
 original triangle. 
 
 When .Tj, x^ and x^ are given, this last equation with the 
 three equations (4) suffice for the determination of d^y d^, d^ 
 and p, and therefore for the position of the point. In fact, by 
 substituting (4) in (5), we have at once 
 
 p{ax^ + hx^ + cx^) = 2 A, 
 
 from which p is found. Then from (4) we have the values of 
 ^j, d^ and d^. 
 
 31 T. Multiplication hy Constant Factors. We may, in- 
 stead of taking x^, x^ and x^ proportional to d^, d^ and d^^ take 
 them proportional to the products obtained by multiplying 
 each distance d by any arbitrary but constant factor. If we 
 take /<!, //j and yUg for such factors, we shall then have 
 
 x^:x^\x^= }A^d^ : }A^d^ : fA^d^, 
 or fA^d, = px^] 
 
 }^A = P^.', \ (6) 
 
 The constant factors /j^, //^ and /j^ being supposed given, 
 the equations (5) and (6) suffice for the determination of 
 d^, d^_, d^ and p. 
 
 318. Definition of Trilinear Co-ordinates. The tri- 
 linear co-ordinates of a point are three numbers propor- 
 tional to the distances of the point from the three sides of a 
 triangle, each distance being, if we choose, first multiplied by 
 any fixed factor. 
 
 The triangle from whose sides the distances are measured 
 is called the fundamental triangle. 
 
 Corollaries from the Definition : 
 
 I. If the trilinear co-ordinates of a point be all multiplied 
 lij the same factor, the position of the point which they repre- 
 sent will 7iot le altered. 
 
 For the position of the point depends only on the mutual 
 
346 MODERN OEOMETRT. 
 
 ratios of its trilinear co-ordinates, which remain unaltered by 
 such multiph'cation. 
 
 11. The points (1), (2) and (3) wliose respective co-ordi- 
 nates are : 
 
 Point (1), {x^, 0, 0), 
 
 Point (2), (0, x^, 0), 
 
 Point (3), (0, 0, x^), 
 
 are the three vertices of the fundamental triangle, no matter 
 what the absolute values of x^, x^, and x^. 
 
 Note. The introduction of the factors // being a mere matter of 
 convenience, the student may ordinarily leave them out of consideration, 
 which is the same as to suppose them unity. Their introduction 
 amounts to supposing that the distances from the different sides of the 
 triangle may be expressed in three different units of length without 
 destroying the truth of our conclusions. 
 
 EXERCISES. 
 
 Prove : 
 
 1. The distances of a point from the three sides of the 
 fundamental triangle cannot all be negative. 
 
 2. Assuming the factors /^j, fx^ and fx^ to all have the same 
 sign, every point whose co-ordinates are all positive or all 
 negative lies within the fundamental triangle. 
 
 3. If the factors fj. are all positive and the trilinear co- 
 ordinates of a point all negative, the value of p must be 
 negative. 
 
 4. The lines from the point {x^, x^, x^) to the three vertices 
 of the triangle divide the internal angles in the respective sine- 
 ratios 
 
 /^,< M,^,' ^^< 
 
 319. Eelation of Trilinear and Rectangular Co-ordinates. 
 Let us suppose the three sides of the fundamental triangle to 
 be given by their general equations in rectangular co-ordinates, 
 as follows: 
 
 Side 1, ax -\- hy -\- c =0; 
 
 Side 2, a'x + h'y -f c' = 0; J- (7) 
 
 Side 3, a"x + l)"y -|- c" = 0. 
 
TRILINEAR CO-ORDINATES. 347 
 
 Then, by §§ 41, 54, if x and y be the rectangular co-ordinates 
 of any point whatever, the expression 
 
 ax -\- by -[- c (a) 
 
 will represent the distance of that point from the side (1) of 
 the triangle, multiplied by the factor Va^ + Jf, Since, by 
 multiplying the equation of side (1) by an appropriate 
 factor, we can give the quantity Va^ + H^ any value we please, 
 we can make it equal to //j. We shall then have, when x and 
 y are the rectangular co-ordinates of a point distant d^ from 
 the side (1), 
 
 //^fZj = ax -\- hy -\- c. 
 
 Thus the equations (6) of § 317 may be replaced by 
 
 ax -{-hi/ -\- c = px^; j 
 
 a'x + yy + c' = px,; [ (8) 
 
 a''x + b"y + c" = px.^. ) 
 
 These equations determine p, x and y when x^, x^ and x^ are 
 given. The values of x and y thus obtained from them are 
 
 _ (yc'' - h"c')x^ + {h"c - lc")x^ + {he' - h'c) x^ 
 ^ ~ (a'y - a"b')x^ + (a''h - ah'')x^ + {ah' - a'h)xj 
 
 - («'V - a'c")x^ + {(^io" - cl"c)x„_ + {a'c - ac') x^^ 
 ^ ~ {a'h" - a"b')x, + {a"h - ab")x, + {ah' - a'h)x^' 
 
 (9) 
 
 which are the expressions for the rectangular co-ordinates of 
 a point {x. y) in terms of its trilinear co-ordinates. 
 
 320, Equation of a Straight Line in Trilinear Co-ordi- 
 nates. 
 
 Theorem. If the trilinear co-ordinates of a point are re- 
 quired to satisfy a linear equation, the locus of the point will 
 he a straight line. 
 
 Proof, Let 
 
 P,X^ + P,^; + i?3^3 = ^ 
 
 be the linear equation which the co-ordinates must satisfy. 
 If we substitute ioxx^, remand x^ their expressions (8) in terms 
 of Cartesian co-ordinates, we readily see that the equation 
 
348 MODERN GEOMETRY, 
 
 will be of the first degree in x and y. It is therefore the 
 equation of a straight line. 
 
 321. Homogeneous Character of Equations. In order 
 that any equation in trilinear co-ordinates may represent a 
 locus, the equation must be homogeneous in terms of such 
 co-ordinates. For, by definition, the position of a point re- 
 mains unaltered when its three co-ordinates are all multi2)lied 
 by any arbitrary common factor p. When we take a homo- 
 geneous equation of the ?ith degree in x^, x^ and x^, such, for 
 example, as (when n = 2) 
 
 ax^x^ + Ix^ — 0, 
 
 and multiply x^, x^ and x^ by p, the result is the same as if 
 we multiplied each member of the equation by p". Hence 
 the relation between x^^ x^ and x^ expressed by the equation 
 remains unaltered. But if we take such an equation as 
 
 ax^x^ + Ix^ -f c = 0, 
 
 and multiply x^, x^ and x^ by c, the result is 
 
 ap^x^x^ + hpx^ -\- c = 0, 
 
 an equation which expresses a different relation from the 
 other. Hence such an equation cannot represent a definite 
 locus so long as the trilinear co-ordinates are used to corre- 
 spond to their definition. 
 
 Correlative of Trilinear Co-ordinates. — 
 Co-ordinates of a Line. 
 
 322. The principle of duality is applicable to all the pre- 
 ceding propositions which express position. We shall there- 
 fore change these propositions into their correlatives. 
 
 Theorem. If three fixed points not in the same straight 
 line le given, the position of a line is completely determined 
 when the mutual ratios of its distances from these points are 
 given. 
 
 Proof. Let the three fixed points be A, B and C. 
 
 Let it be given that the ratio of the distances of a line 
 from the points A and C is m : w. 
 
TRILINEAR CO-ORDINATES. 349 
 
 Let L be any Hue i'ullilling this condition, and let y be 
 the point in which it cuts the line AC. Also let ^la'and 6V 
 
 C B 
 
 be tlie perpendiculars from A and G upon the line L» We 
 then have, by hypothesis, 
 
 Aa' : Cc' :=^m : n. 
 
 Hence, by simil?,r triangles. 
 
 Ay \ Cy =. m: n. 
 
 This last relation completely fixes the position of the 
 point y (§ 279), which therefore remains the same for all lines 
 which satisfy the given condition. That is. 
 
 Every line fulfilling the condition that its distances from 
 two fixed points, A and C, shall he in the ratio m : n, j^asses 
 through that i^oint luhich divides the junction-line AC in the 
 ratio m : n. 
 
 Let it also be given that the distances of the line from the 
 points C and B shall be in the ratio n : p. It must then pass 
 through a point X which divides the junction-line CB in the 
 ratio n -.p. 
 
 If now it be required that the line shall satisfy both con- 
 ditions, it must pass through both the points Xand y, and is 
 therefore completely fixed. 
 
 When both these conditions are satisfied, the distances of 
 the line from BA will be in the ratio p : m, and it will inter- 
 sect AB in some point, dividing AB in the ratio jt? : m. 
 
 The three points then satisfy the proposition of (§ 287), 
 because 
 
 m n P _ -, 
 n ' p ' m ~ 
 
350 MODERN OEOMETRT. 
 
 Proceeding as in the case of the point, if we put 
 i)j, D^, Z^g, the distances of the line from the three points; 
 A J, Ag, Ag, constant factors, 
 we may express the mutual ratios of the distances by three 
 quantities, u^, u^ and tc^, proportional to them. We then have 
 
 W| : tc, : u, = \^D^ : XJ)^ : X^D,; 
 
 A,A ^ ^A ^ ^sA - ^. 
 w, ti^ u, -^' 
 
 X^D^ = Gu^, 
 
 Thus the position of the line is completely fixed by the 
 three quantities Wj, u^ and u^, which are therefore called co- 
 ordinates of the line. We therefore have the definition: 
 
 The trilinear co-ordinates of a line are three numbers 
 proportional to its distances from three fixed points, each 
 distance being first multiplied by any fixed factor. 
 
 The junction-lines of the three points form the funda- 
 mental triangle of reference. 
 
 323. There are therefore two ways of defining the posi- 
 tion of a line, namely: 
 
 1. By the equation of the line. 
 
 2. By the value of its co-ordinates. 
 
 To investigate the relation of these two ways let us con- 
 sider the problem : What are the co-ordinates of the line whose 
 equation is 
 
 rnx^ + nx^ + px^ = 0? (a) 
 
 Let us suppose the indices ^\G 
 
 1, 2 and 3 to refer to the sides 2./^^ \ 
 
 BCf CA and AB respectively. ^^ \ 
 
 Let us first find the point x in^^/_ \ ^ 
 
 which the line (a) intersects ^ ^ 
 
 AB, To do this we put x^ = 0, which gives 
 
 ^1 _ ^ . 
 
 X„ ~ 7?l* 
 
TRILINEAR CO-ORDINATES. 351 
 
 and by substituting for a;, and x^ tlieir expressions (6) in terms 
 of the distances, 
 
 d, _ _ /VL 
 
 d^ M.m ' 
 
 Since d^ and d^ are the distances of x from the sides CA and 
 
 CB, — -- is the sine-ratio in which the line Cx divides the 
 
 angle C (§ 281). To define the point x by the distance-ratio in 
 which it cuts AB, we have the equations 
 
 Bx sin BCx 
 
 
 Cx sinB ' 
 
 
 
 
 Ax sin A Cx 
 
 
 
 
 Cx ~~ sin ^ * 
 
 
 
 Hence 
 
 
 • 
 
 
 Bx sin BCx 
 
 Ax ~ sin A Cx 
 
 sin A _ d^ sin A 
 ' sin B ~ d^ sin B 
 
 _ /j^n sin ^ 
 " //,7?z sin B' 
 
 (b) 
 
 which determines the point of intersection, x. 
 
 By what has been shown, this distance-ratio is the ratio of 
 the two co-ordinates u^ and u^ of the line, multiplied by a 
 factor. In fact, putting D^ and D^ for the distances of the 
 line from A and B respectively, we have 
 
 
 
 Bx 
 
 A 
 
 X 
 
 ,u^ 
 
 
 
 
 
 
 Ax~ 
 
 "A 
 
 ~ X 
 
 ,u. 
 
 
 
 
 Comparing 
 
 this with the equati 
 
 on (b), we 
 
 ) have, 
 
 for the ratio 
 
 of the two line-coordinates. 
 
 
 
 
 
 
 
 
 u,_ 
 
 X^f^^n sin 
 
 A 
 
 n 
 
 K^. 
 
 ! sin 
 
 iA 
 
 
 
 ^1 
 
 A,yUjWsin 
 
 B ■ 
 
 m 
 
 'Kf^. 
 
 sin 
 
 B' 
 
 
 In the same 
 
 1 way, 
 
 
 
 
 
 
 
 
 
 W3 
 
 Kf^sP sin 
 A^yw^^^ sin 
 
 B 
 
 C~ 
 
 ■I 
 n 
 
 
 sin 
 
 sin 
 
 B 
 
 (.0) 
 
 
 Mr_ 
 
 A,/i,msin 
 
 C 
 
 m 
 
 \f^. 
 
 sin 
 
 C 
 
 
 
 M, 
 
 \pi^p sin 
 
 A ~ 
 
 P 
 
 ■ Kf^. 
 
 sin 
 
 A' 
 
 
 Now, it will be remembered that the constant factors X 
 and p. which multiply the distances between the points and 
 
353 MODERN OEOMETRY. 
 
 lines have been left entirely arbitrary. Can we not so deter- 
 mine them that the last fractions in the third members of the 
 above equations shall be unity? This will require the factors 
 to fulfil the conditions 
 
 Ti^fx^ sin B = X^fJi^ sin ^; j 
 
 A^yw, sin C — Ag/Zg sin i^; > {d) 
 
 X^l^t^ sin A = Aj//, sin C. ) 
 
 These three equations are really equivalent to but two, be- 
 cause any one can be deduced from the other two by eliminat- 
 ing the common angle. If we suppose the values of /i to be 
 given, we can determine the mutual ratios of the A's by the 
 equations 
 
 A^ _ /fg sin ^ ^ 
 Ag ~~ /*, sin C^ 
 
 K _ /^3 sin B 
 Ag ~ jLi^ sin C 
 
 Hence the required condition can always be satisfied, and we 
 shall always suppose it satisfied. 
 
 The equations (c) can then be satisfied by putting 
 
 u^ = m X any factor; 
 
 u^ = n X the same factor; 
 
 Wg — p X the same factor. 
 Hence: 
 
 Theorem. If the co-ordinates x^, x^, x^ of a point are 
 considered as variaUes required to satisfy the equation 
 
 mx^ 4- nx^ 4- px^ = 0, 
 
 the point will always lie on the line whose co-ordinates are the 
 constants m, n and p, or their multiples. 
 
 324. Equation of a Point. We have the following 
 theorem, the correlative of the preceding one. 
 
 Theorem. All lines whose co-ordinates ti^, u^ and u^ 
 satisfy a linear equation 
 
 mu^ + nu^ + pu^ = 
 
TRILINEAR CO-ORDINATES. 353 
 
 pass through a point, namely, the point whose co-ordinates are 
 m, n and p. 
 
 This theorem follows immediately from the preceding one, 
 because when a point lies on a certain line, the line passes 
 through that point. Putting both equations in the form 
 
 u^x^ -f ii^x^ + %(,^x^ ^ 0, 
 
 the theorem of § 323 asserts that whenever this equation is 
 satisfied the point {x^, x^, xj lies upon the line (w,, u^, 2iJ. 
 Changing the form but not the essence of the conclusion, we 
 have the theorem that whenever this equation is satisfied the 
 line (n^, w,, ti^) passes through the point (rr,, x^, x^). This 
 result, being true for all values of the six quantities which 
 satisfy the equation, will remain true when we suppose x^, x^ 
 and x^ to remain constant and w,, u^ and Wg to vary; that is, 
 the varying line {u^, u^, u^) will then constantly pass through 
 the fixed point (a:,, x^, x^). 
 
 325. The preceding conclusions may be summed up as 
 follows: 
 
 I. Ifu^, ic^ and u^ are line coordinates, and x^, x^ and x^ 
 are point-coordinates, then, so long as these co-ordinates are 
 unrestricted, they may represent ayiy line and ayiy point what- 
 ever. 
 
 II. If it he required that the line shall pass through the 
 point and the point lie on the line, the co-ordi7iates must satisfy 
 the condition 
 
 u^x^ + u^x^ + u,x^ = 0. 
 
 III. If, in this equation, we suppose the x^s to remain 
 fixed while the u^s vary, the lines represented hy the ti/s tuill 
 all pass through the fixed point represented hy the x's. 
 
 IV. If we suppose the x's to vary while the u's remain con- 
 stant, the points represented hy the x^s will all lie on the fixed 
 line represented hy the it's. 
 
 336. For brevity of writing we may use a single letter to 
 represent the combination of three co-ordinates of a point or 
 line. Then the expression **the point {p)" will mean the 
 point whose co-ordinates are />„ p^ and p^. 
 
354 
 
 MODERN OEOMETRT. 
 
 Theorem. If {x) and (y) are a?iy two points^ and if, 
 with any factor A, we form the quantities 
 
 ^1 — ^1 4" ^Vi^ ) 
 
 z^ = x^-\- Xy„ V {a) 
 
 2^3 = ^3 + ^Vz^ ' 
 the point (z) luill lie on the li^ie joining the points (:f) and {y). 
 Proof. Let (p) be the line joining the points (a:)' and (y). 
 Because the line {p) passes through the point {x), we have 
 
 Because {p) passes through the point {y), we also have 
 
 p,y^ + P.y. + ^^3^3 = 0. 
 Multiplying this equation by X and adding it to the other, 
 
 pM. + ^y.) + pX^^. + ^.) + pA^. + ^^3) = 0, 
 
 or ;?,2j + ^2^2 + i^3^» = ^- . ■ 
 
 Therefore the point {z) is on the line (p). Q. E. D. 
 
 Cor. Any point (2;) whose co-ordinates satisfy the con- 
 ditions 
 
 z^ = Xx^ + /uy„ 
 
 z, = Xx^ + My^, 
 
 z, = ^x, + >"2/3» 
 
 lies on the same straight line with the points (x) and (//), what- 
 ever be the factors X and yw.* 
 
 327. Problem. Having the four 2')oints in a right line, 
 {x), (y), {x-{-Xy), {x-^rX'y), 
 
 it is required to determi7ie their a7iharrnonic ratio. 
 
 We must first, instead 
 of the trilinear co-ordi- v^ 
 
 nates, take the actual re- 
 duced distances of the 
 points from the sides of A" 
 the triangle. Let us then suppose 
 
 * These coefficients A and f.i must, of course, not be confounded with 
 the factors used in g§ 317, 322. 
 
TRILINEAR CO-ORDINATES. 355 
 
 AB, any side of tlie fundamental triangle; 
 
 z E the 
 
 point {x-\-\y)', 
 
 ax, by. 
 
 cz ^ p, q 
 
 and r, the distances of x, y and z from 
 
 AB. 
 
 
 
 We then have for 
 
 the distance-ratio of z, with respect to 
 
 X and y, 
 
 
 xz r — p 
 yz r - q "^ 
 
 This gives 
 
 
 j.-J> - ^9 
 
 1-M 
 
 We have three equations of this form, corresponding to 
 the three sides of the fundamental triangle, in which p, q 
 and r have the respective indices 1, 2 and 3. We may write 
 them: 
 
 (1 - M)r^ =p, - MQr; 
 
 (1 ~ M)r, = p, - Mq,; 
 
 {l - M}r, = p, - Mq^' 
 
 The trilinear co-ordinates a;,, x^ and x^, proportional to 
 
 jOj, p^ and ^3, are formed by dividing these last quantities by 
 
 a common factor = p. Let <j be the common factor for q. 
 
 Then these equations become, by substitution and reduction. 
 
 
 '-^'■ 
 
 = ^, • 
 
 
 
 
 '^'■ 
 
 = ^, • 
 
 
 
 . 
 
 '-^" 
 
 = ^3 ■ 
 
 
 
 These equations 
 
 become identical 
 
 with {a) by putting 
 
 1- 
 
 z = 
 
 P 
 
 '' P 
 
 = K 
 
 or 
 
 "-^ 
 
 But yu is the distance-ratio of the point {z) = (a: -f ^v) with 
 respect to the points {x) and (?/). Hence, 
 
 When from two poijits, (x) and (y), we form the third, 
 (x -\- \y), in the same straight line, the distance-ratio of the 
 
356 MODERN GEOMETRY. 
 
 third tuith respect to the other two is equal to X multijMed hj 
 
 a factor, — — , depending ui^on the absolute values of the tri- 
 
 lincnr co-ordinates. 
 
 Since, from the nature of trilinear co-ordinates, this factor 
 remains indeterminate, the distance-ratio also remains inde- 
 terminate. But if we also take the distance-ratio of a fourth 
 point, (x + ^'y), and then form the anharmonic ratio, this 
 factor will divide out, and we shall have 
 
 Anh. ratio = jj. 
 
 Hence the anharmonic ratio of the four points (x), (y), 
 
 (x + Xy), {x -\- X'y) is -77, which solves the problem. 
 
 Cor. Harmonic Points. Since four harmonic points are 
 such whose anharmonic ratio is — 1, we must then have 
 A' = — /I. We therefore conclude that if we have any four 
 points capable of being expressed in the form 
 
 W. {y), (^ + '^y)y {^ - ^)y 
 
 the last pair will divide harmonically the segment contained 
 by the first pair. 
 
OOUESE OF EEADING IN GEOMETET. 
 
 The following classified list of books is prepared for the 
 use of students who wish to continue the study of the subject: 
 
 I. MODERN SYNTHETIC GEOMETRY. 
 
 Chasles, Traite de Geomelrie SupSrieure (546 pages 8vo), is noted for 
 
 its elegance of treatment. It is principally confined to tlie geometry 
 
 of lines and circles. The subject is continued in 
 Chasles. Traite des Sections Coniques, Premiere Partie^{no second part 
 
 publislied), 
 TowNSEND, Modern Geometry of the Point, Line and Circle (2 vols. ; 
 
 Dublin, 1863), covers ground similar to the first work of Chasles, 
 
 but is more elementary. 
 Steener, Vorlesungen iXber Synthetische Geometrie, is a very extended 
 
 treatise, but lacks the clear presentation of Chasles. 
 
 II. PLANE ANALYTIC GEOMETRY. 
 
 Salmon's Conic Sections and his Higher Plane Curves treat the subject 
 with the clearness and simplicity which characterize the works of 
 that author. 
 
 Clebsch, Vorlesungen ilher Geometrie, of which there is a French 
 translation, is the most complete single treatise on the higher 
 branches of modern geometry now at the command of the student. 
 
 III. ANALYTIC GEOMETRY OF THREE DIMENSIONS. 
 
 Salmon, Analytic Geometry of Three Dimensions, is the most extended 
 treatise in English. It presupposes a knowledge of the elements 
 of modern algebra, such as can readily be derived from his treatise 
 on that subject. 
 
 Frost, Solid Geometry, is less extended than Salmon's treatise, but 
 written more in the style of a text-book. 
 
 Aldis, Solid Geometry (223 pages 12mo), is an excellent little elementary 
 text-book, with numerous exercises. 
 
 Hesse, Voi'lesungen iiber Analytische Geormtrie des Raumes, is a Ger- 
 man work of 450 pages, noted for its elegance of treatment. 
 
UNIVERSITY OF CALIFORNIA LIBRARY 
 BERKELEY 
 
 Return to desk from which borrowed. 
 This book is DUE on the last date stamped below. 
 
 JAN 3 1945 
 
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 TOY 7 1969 9 4 
 
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 OCT 2 8 1977 
 
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