Irving Stringham f TV <>J NE WCOMB' S Mathematical Course, I. SCHOOL COURSE. Algebra for Schools, $1.20 Key to Algebra for Schools, .... 1.20 Plane Geometry and Trigonometry, with Tables, 1.40 The Essentials of Trigonometry, .... 1.25 II. COLLEGE COURSE. Algebra for Colleges, $1.60 Key to Algebra for Colleges, .... 1.60 Elements of Geometry, 1.50 Plane and Spherical Trigonometry, with Tables, 2.00 Trigonometry (separate), 1.50 Tables (separate), 1.40 Elements of Analytic Geometry, . . . 1.50 Calculus (in preparation), Astronomy (Newcomb and Holden,) . . . 2.50 The same, Briefer Course, .... 1.40 HENRY HOLT & CO., Publishers. New York. 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. i^u-^- cJUZLaji. o-^ '^^^ '^^^^^''^- '^^^^^-^ ''^^^^'■^H^ r Cct^t /3 ,<t.^ij^^^r-»-i^<l^<2 «. C't-^'-ir^oL l^^r^i^y-CU^ Cct.cyC^-^^ 4y<o /^ yi^'piLc ^£. ^> / r^j /ae^ ao^ oc hhr (p^J ^-T-= / r-^; ^^rU^^^ Jc ^^i^ _ A > t- TF. JSL'^ /--^ ^ y A- 7^ /- JZ /--€^ (^^ •'Z- j(Z>- _ ^ ££-€ ^^ ^ ^^^^-^^^-felv (r/--^V^^^- ■;^ 2/. ^ ./. ^^7^/,^ ^jV J^lyXjC. o^^r<~ ^ - f-^ -^^C ^..^ r ':^r^^JLOC t ,«^ S^^uJ F' C p C "^ — ^^ — -^ *" ■ — - — ^ - - — ^- =^ — — ^ Ctfi^ 5^*x^F/vP s^pp/v^ ^>^r'/vP^ »c*^/"7^A/ 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 9 270cf48Jr { 2Nov*5bHF llApr'5lFM \6l^\ 60 WW TOY 7 1969 9 4 *-.OAN DEPT» OCT 2 8 1977 «EC CIR.NOV "2 11 LD 21-100m-9,'47(A5702sl6)476 800557 :^iv53^ r^\ UNIVERSITY OF CALIFORNIA LIBRARY U. C. BERKELEY LIBRARIES CDbl35DflbD