TKEATISE ON THE PRINCIPLES AND APPLICATIONS OF ANALYTIC GEOMETRY. BY HENRY T. EDDY, C.E., PH.D., PROFESSOR OF MATHEMATICS AND ASTRONOMY IN THE UNIVERSITY OF CINCINNATI. PHILADELPHIA: COWPERTHWAIT & COMPANY. 1874. Entered according to Act of Congress, in the year 187 U, by HENRY T. EDDY, /Z^73J In the Office of the Librarian of Congress, at Washington. WESTCOTT & THOMSON, EDMUND DEACON, Stereotype and Electrotypers, Phila. Printer, Phila. PREFACE. THE following treatise, designed as a text-book upon Analytic Geometry, has been written with the most practical ends in view, and is intended to meet the wants of classes in Scientific and Technological Schools, Colleges and Universi- ties. While the needs of the student of Mechanics, Astronomy and Civil Engineering have never been forgotten, it has been found possible to so select the material and to put it in such shape as to adapt the work- to the student who pursues the subject merely as a part of a liberal education. The prime difficulty the ordinary student meets in the 'study of analytic geometry is in the use of variables, since with these he has had no previous acquaintance. No pains has been spared to make the introduction to their use clear and free from all other complexities. To this end a thorough knowledge of co-ordinates has been first secured by the study of the relations of points, the transformation of co-ordinates, etc. Again, the entire subject of the general relation of constant and variable quantities is postponed to Chapter V, at which point the student will have attained a sufficient acquaintance with the processes and notation peculiar to analytic geometry to grasp the ideas advanced and use them in after work. To secure an accurate knowledge of the meaning of the general equations, it is essential that the student should solve numerous numerical examples. They should be illustrations, and of such simple character as to be readily solved by any one who understands the preceding text. Such are the examples interspersed through the work, which should in no case be omitted. Indeed, if the class is numerous, the teacher is advised to largely increase the number of examples as class-room work by substituting other numbers than those used, and giving each example to a sufficient number of different computers to ensure correct results. The Exercises are much more difficult than the examples, and have two objects in view : first, as original work for the more ambitious students ; and secondly, as results to be referred to in the students' subsequent studies. They may be omitted by the ordinary student. The great difficulty which the teacher experiences is not usually that the 4 PREFACE. student cannot be made to apprehend the true import of the demonstrations, but it is this that he afterward fails to recall the necessary equations and their significance. To assist the teacher in this vital point, the statement of each theorem is in a form to be memorized, and contains some important equation and its signifi- cation. The importance of acquiring a perfect familiarity with these statements in algebraic language instead of ordinary language cannot be too strongly emphasized. It has been found by the best teachers that ten or fifteen minutes during every recitation hour should be spent in reciting from memory the state- ments of all theorems previously studied. The form of notation adopted is thoroughly systematized, and prepares the student to read with ease the great modern writers upon analytic geometry. The marked value of the angular notation used is a sufficient recommendation for its adoption. For it I am happy to acknowledge my indebtedness to Prof. J. M. Peirce, of Harvard University, from whose works it is borrowed. The one great defect of text-books upon analytic geometry is the omission of general principles. It appears to be assumed that an acquaintance with its ordinary processes gives the student a knowledge of its principles. This is far from being a correct assumption, as an examination of the general princi- ples stated and demonstrated in Chapter V will abundantly show. The general discussion of curves and their singularities by means of their approximate equations a method due to the genius of Newton is here for the first time rendered accessible to the ordinary student, and it is thought that it will serve a most useful purpose by putting into his hands an instrument of research of practical value whose power compares favorably with the resources of the differential calculus. No attempt has been made in the last chapters to follow the beaten track of previous text-books, but rather to select matter respecting spirals, etc., of the greatest use to the student. The book will be found to be suited to the wants of classes taking either a longer or shorter course by the various omissions indicated in the course of it. I take this opportunity to express my thanks to Prof. James Edward Oliver, of Cornell University, for many happy suggestions. I am especially indebted to Prof. E. W. Hyde, formerly of Cornell University, who has with signal ability and fidelity assisted me in preparing the book for the press. In particular, the Examples were, almost without exception, made by him. Part Second, upon Solid Geometry, is in preparation, and will be issued at as early a date as circumstances may permit. ---HENRY T. EDDY. PRINCETON, NEW JERSEY, August 15, 1874. CONTENTS. PAGE ABBREVIATIONS AND SIGNS. GREEK ALPHABET 7 INTRODUCTION 8 CHAPTER I. Co-ordinates. Systems of Co-ordinates , 9 Positive and Negative. Distance and Angle 11 CHAPTER II. The Point. Cartesian Co-ordinates 14 Relations of Points 18 Polar Co-ordinates 22 Projections 28 Transformation of Co-ordinates 30 CHAPTER III. The Right Line. The Right Line in Cartesian Co-ordinates 39 Relations of two or more Right Lines 47 Perpendicular and Direction Cosines 54 , Right Line in Polar Co-ordinates GO Exercises 62 CHAPTER IV. The Circle. Tangent and Normal of any Curve 64 The Circle in Rectangular Co-ordinates 65 The Tangent and Normal Line 09 Centre and Axes of Similitude 73 Polar Equation '.. 76 Exercises 76 CHAPTER V. Equations and Loci. Definitions 79 Properties of Equations and their Constituents 81 Symmetry , 91 5 CONTENTS. CHAPTER VI. The Parabola. PAOB Definition of the Conic Sections 93 The Parabola in Rectangulars 94 The Tangent, Polar and Normal Lines 96 The Parabola in Polar Co-ordinates 105 CHAPTER VII. The Hyperbola and Ellipse. The Hyperbola and Ellipse in Rectangulars 108 Tangent, Polar and Normal Lines 118 Conjugate Diameters and Eccentric Angle 128 Asymptotes of the Hyperbola... 141 Hyperbola and Ellipse in Polar Co-ordinates.... 144 CHAPTER VIII. General Equation of the Second Degree. Co-ordinates of Centre and Direction of Axes 146 Invariants and Criteria 150 CHAPTER IX. Curves of the Third and Fourth Degrees. Curves of the Third Degree 153 Curves of the Fourth Degree 159 CHAPTER X. Higher Algebraic Curves. Parabolic and Hyperbolic Curves 163 Exponential Polygon and Approximate Equations 167 Multiple Points 178 CHAPTER XI. Transcendental Curves. The Cycloids 180 Trigonometric and Auxiliary Curves 185 CHAPTER XII. Spirals and Polar Curves. Parabolic and Hyperbolic Spirals 196 Exponential Polygon and Approximate Equations 199 Exercises... .. 200 Abbreviations and Signs. E. G. is used to introduce an illustrative example. N. U. is used to introduce some useful notation or convention adopted. A period may signify multiplied by, and a colon may signify divided by. T = 8.14159 is the semi-circumference of the circle whose radius is unity. . . signifies therefore, and i. e. signifies that is. OPQ may signify the angle OPQ, etc. The dagger (f ) signifies that the proposition to which it is affixed may be omitted if desirable. (x, y) signifies some unknown function of P*Pr X' in which />, p m and /> r have any directions whatever. Therefore the equation a + ^ + + +^ 1 =^ 1 expresses the final angle between x and a line turning from x successively to a /? 7- and /> L . It is to be noticed that all parallels have the same direction, so that any line has the same direction as a parallel to it through 0; from which it follows that the sum of the exterior angles of any convex polygon is equal to 360, as is also proved in geometry. It is also to be noticed that any multiple of 27r = 360 may be neglected, since a complete revolution does not change the position of a line. Also that ~~ * = 180. JC 13. Since y x = *, in which x and y have any directions what- ever, we have by trigonometry : tan | = tan (-*) = - tan* cot | = cot (-*)= -cot* sec | = sec (-*) = + sec * cosec y cosec ?. = cosec ^^ CHAPTER II. THE POINT. CARTESIAN CO-ORDINATES. 14. Abscissa, or x. The distance OA l =B l P l is called the abscissa of P 1; or the x co-ordinate of P x , or simply the # of PI. A, / When this co-ordinate is measured to the right of 0, it is usually reckoned positive, and when measured to the left, negative. 15. Ordinate, or y. The distance OB l = A l P l is called the ordinate of P lt or the y co-ordinate, or simply the y of P l4 It is usual to consider y positive above, and negative below 0; e. g., for the point P 3 , x and y are both negative. 16. Axes. The lines OX and Fare called the axes of reference, or the co-ordinate axes, or simply the axes of # and y. The axis of re is usually taken horizontal. 14 ORIGIN AND AXES. 15 17. Origin. The point 0, in which the axes of x and y inter- sect, is called the origin of co-ordinates, or simply the origin. + + The angle XOYis called the first angle. " " XOY " " second" " " XOY " " third " + - " " XOY " " fourth " For a point situated in the first angle, x is + and y is +. in the second angle, x is and y is +. in the third angle, x is and y is . in the fourth angle, x is + and y is . 18. Oblique Axes. When the angle between the axes of x and y is not 90, the system is called oblique. + 4* It will be convenient to use at = XO Fto denote this angle. Rectangular Axes. When o = 90, the system is called rectangular. 19. N. B. We shall use x and y to denote the co-ordinates of any point P with reference to the axes OX and Y, We shall use x 1 and 2/ to denote the co-ordinates of any point P with reference to other axes, OX' and O Z', and a;" and y" with reference to 0"X" and 0" Y", etc., etc. We shall also for convenience use x l and y v to denote co-ordinates of some particular point jP 1? i. e., x l and ^ are particular values of x and y. Also # 2 and y 2 , for the point jP 2 > are other particular values of x and y, etc., etc. Similarly, a?/ and y/, or #/ and y/ are particular values of of and /. 16 THE POINT. Proposition 1. 20. TJieorem.The equations x = Xi and y=y\ represent a point ; in which x and y may be the bilinear co-ordinates of any point, and Xi and y l are their values for this particular point. For a single point has position only, and its position is com- pletely determined by these equations. N.JB. We shall frequently speak of the point (x^y^) t or ( 2, 3), etc., meaning the point whose co-ordinates are x = x^ y y\, or# = #, 2/ = #, etc. 21. Corollary. The equations x = and y = represent the origin. 22. Examples. Locate the points represented by the following equations, both in rectangular and oblique co-ordinates, using first \ in., then i in., as the unit of measure. (1.) * = *, y= 6 . (2.) x = -l, y = 3.5. (5.) (6.) (7.) (8.) TRANSFORMATION. Proposition 2. 23. Theorem. The equations I "' and y = yj- Substituting in the previous equations, we have x f = x ' -f a?, and y' = y'^y, .'. x = x f xj, and y = y' y '. These axes may be either oblique or rectangular. The above change of axes of reference is a case of transformation of co-ordinates. This change will evidently not affect the relations of the point to any other points or lines than the origin and axes. 18 THE POINT. 25. Examples. Construct the axes and points in the following examples (see Art. 18) : (4.) y == 9 / .J (j> n / G) fy* Q X ~ fa X O, y ^, //0 ^> y = 5, y = , 2/o = -4, ij<> = 2, a> = 45. a> = 90. a> = 120. o> = ^. (5.) The three vertices of a triangle are the points, (2, 1\ (3, 4), ( 3, 2), and the co-ordinates of the new origin are X Q = 5, y = 4 ; construct the triangle, and find the new co-ordinates of the vertices, when 10 = 90 : also when at = 120. Proposition 3. 26. Theorem. The equation expresses the distance between two points ; in which r is the distance, and (xi,yi) and (z 2 , y 2 ) are the rectangular co-ordinates of the points. Since the square on the hypote- nuse is equal to the sura of the squares on the two sides of a right- angled triangle, the equation is evidently true from inspection of the figure. For P^= IFD + P^D = (OC- OA}+ (CP 2 - CD)] 27. Cor. If x l = 0, and y^ = (Art. 21), then r =|/^ 2 2 + 2/ 2 2 . It is to be noticed that this proposition is general, though proved only in the first angle. DISTANCE. AREA. 19 28. Schol. The expression r = -\/(x t xtf + (y 2 yO 2 , when transformed to new parallel axes by the equations (Art. 23) is r = i/(x 2 ' x^y + (y/ y/) 2 . For a?, - a* = [a* + ar a ' - (x 9 + a?/)] =* 2 ' - a?/, and similarly y 2 yi = y/ y/. It is to be noticed this transformation does not affect the relation of the points to each other. 29. Examples. Find the distances between the following points : (t) (3, I), and (8,5). Ans. 7.81. (2.) (2, 4\ and (1, 2). (&) (3,7), and ( 3, 5). (4.) Refer the points in Ex. 1 to new parallel axes, the co-ordinates of the new origin being x = 6, y = 2. (5). In Ex. 3 refer to new parallel axes, making x = 3 and y = 7. 30. Exercise. Prove that when the axes are oblique, r = i/(# 2 xtf -I- (y 3 yO 2 + 2 (x 2 xj (y 2 yj cos w. Proposition 4.f 31. Theorem. The equation expresses twice the area of the triangle whose vertices are the three points, of which the rectangular co-ordinates are (#1, 2/i), (*, 2/2), (a?s 2/s)> <*< 0/ which the area is t. f All propositions or other divisions of the book marked ( f ) may be omitted on first reading. 20 THE POINT. For, t = l j r a. 2 b. 2 ) -fe-^i) (y 3 +yi) tti 0,3 0,2 X And by multiplying out and canceling we obtain, 32. Cor. In the same manner we can obtain the area of any poly- gon. The area of a quadrilateral = q is given by the equation, * # 33. Schol. 1. If the preceding points be referred to new axes parallel to the primitive, we shall have (Art. 23), yi (*,' - ti) + yi (?l - x!) + yl W - x{) = to, since all the terms that contain X Q and y will cancel. The relation between the points is unchanged by the transformation. 34. Schol. 2. It is useful to notice the cyclic symmetry of the above ex- pression that is, that the subscripts follow around in the same cyclic order, viz., 123, 231, 312. 35. Examples. Find the area of the surfaces inclosed by right lines joining the points whose co-ordinates are given in the following examples : (1.) (2,3), (4,1) and (5, 6). Ans.6. (2.) (-5,1), (-2, -3), (4,6). An*. 26$. (3.) (-4,1), (6,3), &-1), (-3, -2). Am. 51. THREE POINTS ON A LINE. 21 36. Exercise. Prove that when the axes are oblique, [y\ fe #3) + 2/2 (^s #0 + 3/3 (x\ #2)] sin w Proposition 5. 37. Theorem. Tlie equations n, x, __ express the position of a point dividing the distance between two points in a given ratio; in which n\ : n 2 is the ratio, and (x, y} are the co-ordinates of the point dividing the distance between (x^y^) and (x t ,y t ) in the given ratio. For,letOa=z, Oa l =x 1 and By similar triangles we have bid CD : : b,P : Pb 2 , or Similarly y = J ^ X a a 38. Schol. 2. If the preceding points be referred to new axes paral lel to the primitive, since (Art. 23), x x v = x + x' (x + Xi) = x f or/, etc. ; . . of Xi : x 2 ' x 1 : : n^ : n^ ; d ni + + This transformation does not change the mutual relation of the three points. 22 THE POINT. 39. Schol. 2. Were the point not situated between (x lt ?A) and (#2, y*)> but on either sidb of these points, n? would be negative, and the equations would become d 40. Examples. (1:) Find the co-ordinates of the point which divides in the proportion of 3 to 5 the line joining the points and > Ans. < 3 8 \y=i. Given a line joining two points (5, #) and (8, 7), to find the co-ordinates of a point on the line not between the given points, which shall divide the line in the proportion of 1 to 7. 41. Exercise. Prove this proposition also as a case of Prop. 4, when t = 0. POLAR CO-ORDINATES. 42. Radius Vector. The distance OP is called the radius vector, or p co-ordinate, or simply the p of P. It is usual to consider p as positive when there is no reason for taking it negative. 43. Variable Angle. The angle XOP is called the variable angle, or 6 co-ordinate, or simply the 6 of P. It is usual to measure this angle from XO around in a direction opposite to the motion of the hands of a watch for positive rotation, and with the hands for negative rotation. It is often convenient to denote the angle between x and /?, as p =0, which must be carefully distinguished from a ratio or fraction. (Art. 12.) POLAR CO-ORDINATES. 23 44. Pole. The point in which all the radii vectores inter- sect is called the pole. The pole is the origin of distance. Initial Line. The line OX from which the variable angle is measured is called the initial line. The initial line is the origin of direction. 45. N. B. "We shall use p and to denote general values of the polar co-ordinates referring to a primitive pole and initial line, and p' and tf referring to a new pole and initial line. We shall also use f) l and #i, PZ and 2 , etc., to denote restricted values. (Compare Art. 19.) Proposition 6. 46. Theorem. The equations p=p l and = 0! represent a point; in which p and may be the polar co- ordinates of any point, and p lt O l , are their values for this particular point. For, a single point has position only, and its position is com- pletely determined by these equations. N. U. We shall use (p l} 0J, or (5, -), etc., to indicate the point whose co-ordinates are p=pi, = 0^ or p = 5, = ^, etc. 47. Cor. The equation p = represents the pole, and the equa- tion = 0, or = -, or = 2 TT, etc., represents the initial line. 48. Examples. Locate the points whose co-ordinates are given below. (1.) p =2, = 1. (2.) P = -l, = 45. e (3.) P= , = 2^. 24 THE POINT. Proposition 7. 49. Theorem. The equation is that by which any point is referred to a new initial line through the pole; in which 0o is the angle between the new and the primitive initial line. From inspection of the figure, P X ' _L P >r> A ft -4- ft' Observe that the transformation of this one of the polar co-ordinates of any point, will not change its relations to anything except the initial line itself. Proposition 8. 50. Ttieorem.The equation f 1/Y'i 2 + P/ ~ Mf'i ('2 cos (#1 #2) expresses the distance between two points ; in which r is the distance between the two points whose polar co-ordi- nates are (/>i, 0i) and G2, #2). If a } 02 = r, Oai = p l} Oa 2 = p 2) i and * = 6 l 2 , we have, by trig., (Compare Art. 30.) 51. Cor. If one of the points, say a 2 , be at the origin, r = p. 52. Schol. If the initial line be changed to coincide with p 2t i.e., = 2 , .'. 0i - 2 = + */ - ^o, -/>!/>, cos AREA OF TRIANGLE. 25 53. Examples. Find the distances between the points whose co- ordinates are as follows : (!) (2.) (3.) (4.) Am. r = 6.08 t =8,0 = 350, and p = 6, = 80. Ans. r = 5. 9 ' Ans. r = 10. Ans. r = 8.72 Proposition 9.f 54. Tlieorem.The equation p 1 p 2 sin (0i 2 ) + p 2 ps sin (0 2 3 ) + p 3 p l sin (0 3 0j) = expresses twice the area of any triangle, when t is its area, and the polar co-ordinates of its vertices are (P\, 0\), (f>*, 0*}, and G3, 0s). The triangle 123 is equal to 012 + 023 + 031, and by trig, the area of 012 is 01X02X sin 102 p, p 2 sin (0! - 2 ) and similarly for the other triangles, . . ptp 2 sin (0! - 2 ) + p 2 ps sin (0 2 - 3 ) + p 3 p l sin (0 3 - 0^ = 2t. This expression is equally true when the pole is not within the triangle. Observe the cyclic Bymmetry in the above expression. 26 THE POINT. 55. Cor. In the same manner we may derive an expression for the area of any polygon. 56. Examples. Find the areas included within right lines joining the points whose co-ordinates are given below. (1.) (5, 10), (2, 100), (3, 200). Ans. t = 9.26 < 2 -) * * ' Proposition 57. TJieorem.The equation 2 - (pf - pf) (pf - pf) + rfpf (r? + n 2 - rf) - rf C 3 2 - P?) (!>? ~ P/) + rf tf (r? + rf - rf) = rf r} r," expresses the relation between the length of the sides and diagonals in any quadrilateral i. e., the six distances be- tween any four points. For identically, (03-dJ + (Oi-0 2 ) = 3 -0 2 , or + /9 = r , if 0,-d l = a ) flx-02^/3 and 03-02 = ;- By trig, sin a cos /3 + cos a sin /9 = sin f Squaring, sin 2 a cos 2 /9 + 2 sin a sin /? cos cos /9 + cos 2 a sin 2 /? = sin 2 / . . (-? cos 2 ) cos 2 j9 + ^ sin a sin /9 cos a cos /9 + cos 2 a sin 2 ft==l cos 2 ^, . * . cos 2 a (cos 2 p + sin 2 /?) + cos 2 /9 4-cos 2 y 2 cos a cos /9 (cos a cos /? sin a sin /9) ==^, . ' . cos 2 a + cos 2 /9 + cos 2 p 2 cos a cos ^ cos f = l ..... (a.) FOUR POINTS. 27 By Art. 50, ll'2 o x Substituting these values in eq. (a.} } we have the above result. Proposition H. 58. Theorem. The equation AP . QB sin AOP sin QOB AQ . = a constant expresses the relation of the four distances between the four points in which any line intersects a pencil of four rays. For, if p = length of the perpendicular let fall from upon AB, we obtain the following expressions for double areas of triangles. p. AP = OA. OP sin AOP . . p. QB = OQ . OB sin QOB . . p.AQ = OA . OQsmAOQ p.PB = OP . OBsmPOB (d.) o The product of equations (a.) and (6.), divided by the product of equations (c.) and (d.), gives the above result, which is the same for every line intersecting the pencil. 28 THE POINT. 59. Scfiol. AP. QB : AQ . PB is called the anharmonic ratio of the pencil 0-APQB. The ratios AP . QB : AB . PQ and AB.QP-AQ.PB are also of constant value, as may be proved in a similar manner. PROJECTIONS. 60. Projection of a Point. The foot of a perpendicular let fall from any point upon a line is the orthogonal (i.e., rectangular) projection of the point upon the line. Similarly the oblique pro- jection of a point may be obtained, but the orthogonal projection will always be understood unless otherwise stated. 61. Projection of a Distance. The distance between the pro- jections of two points is the projection of the distance between the points. Proposition 12. 62. Theorem. The equations x 2 x 1 =r cos , and y 2 y\ = r sin T x express the projection of the distance between two points upon the rectangular axes of x and y; in ivhich r is the distance between the two points For, if im = x 2 x l , and71? = r, by trig. x z x 1 = r cos x 2 Xi r cos T x Similarly y 2 - y l = r cos (90 Q - ^ .*. by trig. y 2 y 1 = rsin![/ ^ ^2 -X The given distance is the hypotenuse of a right-angled triangle, and its projections on the axes of x and y are respectively parallel and equal to the base and perpendicular. PROJECTIONS. 29 63. Cor. If the point 1 coincides with 0, x- 2 = r cos r , and y- 2 = r sin r . 64. Example. Find the projections on the axes of x and y of the distance between two points, when = , and;* = JC Ans. # 2 #1 = , and y^ y l = 3. Proposition 13. 65. Theorem. The equations TI cos !} = r 2 cos !? -f r B cos sn r 2 sin sn express the fact that the projection of one side of any triangle upon any lines, as upon the rectangular axes of x and y, is equal to the sum of the projections of the two remaining sides upon the same line ; in which r lt r t and r 3 are the three sides of the triangle. For, Also, = 02 a : + a\ a 3 cos = r 2 cos T + r 3 cos & s = &2&i + &i& 8 ; (Art. 9.) sin = r 2 sin 2 + r 3 sin J, 3 X 66. Cor. Since r 2 might be the third side of a new triangle, etc., it follows that the sum of the projections upon the axis of ar, of any broken lines leading from 2 to 8, is equal to the projection of the distance 2 8 upon the axis of x. By the word sum is to be understood algebraic sum, since the projections of any points falling without a^ a$, would give us negative distances. (Art. 9.) 30 THE POINT. 67. Example. Show the truth of the preceding proposition numeri- cally by applying the formula to the following data. Co-ordinates of (!),* = $, y = 2;of(2),x = 6, y = l; of (3), a? = 0, y = 6; r\_ 25" = = 161 34'. TRANSFORMATION. 68. The Transformation of the co-ordinates of any point is the reference of the point to a new system of co-ordinates. / N. B. Review Arts. 12, IS, 18 and 19. Proposition 14. 69. Theorem. The equations x sin * = x' sin ^' + y' sin ^ y sin y x = a;' sin *' + y ' sin % are the equations of transformation of any point from a primitive system of oblique bilinear co-ordinates to a new system, also oblique, the origin remaining the same; in which x and y are the primitive co-ordinates of the point, and x' and \f the new co-ordinates of the same point. Project the figure OAPA' upon PD drawn perpendicular to the axis of x. Then, by Art. 65, Similarly, if a perpendicular be let fall upon the axis of y, and we can prove that x sin x y = x' sin *' + y' sin jj'. Notice the symmetry of these formulae. A D X TRANSFORMA T10N. 31 70. Examples. (1.) Refer the point x = 3,y = 5, when ^ = 120, to new axes in which ^/ = 0, and * =30, the origin remaining the same. Ans. *' - '0.577, y' == 4.0^ (2.) Refer the point x = 2, y = 5, when ^ = 0, to new axes such that y x 120, and y x , =80, the origin remaining the same. Ans. x' = - 2.64, 2/ = - 8.045 Proposition IS. 71. TJieorem.The equations x = x' cos 4- y' cos & *c> */ *O y =y' sin ^ + x 1 sin * transform from rectangular to oblique axes, the origin remaining the same. Y /r By projections x = x' cos x + y' cos ^ 6 ^ / 72. Cor. If the axes of x and x' coincide, then ?' = 0, .'. ^ If the axes of y and y coincide, then y,=o, = a; cos 32 THE POINT. 73. Examples. (1.) Eefer the points (5, ), (- 4, 0) when = 00, to new axes for which ^ = $0, and #' = 6(9. x x Ans. For first point, x' = 0.196, tf = 5.6(5 For second point, x' = 8.93, y' = 746 (2.) Refer the point (3, ) when ^ = 00 to new axes for which x' Ans. x' = - 0.732, tf = 6.175 74. Exercise. Prove the equations of Art. 71 directly from Art. 69. Proposition 16. 75. Theorem. The equations x sin x y = x f sin *' + y' cos *' y sin y = x f sin * + y' cos *' ^ *. \<4 >/.. " 77. Cor. 2. If the axes of y and ?/ coincide, ( ^ / v/ v> " , then = 90. y sin = a;' sin cos 78. Example. Refer the point (3, 6) when ^ = 60 to new rec- tangular axes, such that ^ = 70. Ans. x' = 1.719, ?/ = 4-^03 79. Exercise. Prove the equations of Art. 75 from the figure directly. Proposition IV. 80. Theorem. The equations x = x' cos -J y' sin ^ yx' sin ~ +?/ cos ~ transform from rectangular tj new rectangular axes, the origin being unchanged. For, in the equations of Art. 69, let y = y ' then = += . . sin |' = sin (^ + 90\ = cos Also f^^^-^ .'. sin ^= sin (^-00)= -c Moreover, sin ^ = sin * = 7, and sin'= sin . THE POINT. Substituting these values in the equations of Art. 69, we have, x x f cos x x y ' sin x x y = x f sin *' + y' cos *'. This is the most common transformation, and may also be written x = x f cos 6 y' sin 6 y = #' sin 6 + y' cos 0, 6 being the angle through which the rectangular axes are revolved. 81. Example. Refer the point (5, 2) to new rectangular axes so that we shall have 6 = 30. Ans. x' = 3.33, y'^ 4. 82. Exercise. Prove the equations of Art. 80 directly from the figure. Proposition 18. 83. Theorem. The equations Y = n fWS P ti n ro P n air P JU &J OUo ~ij U tJ L/Uo fJ Sill transform from rectangular to polar co-ordinates, the pole being at the origin, and the axis of x the initial line. For, y x = 90, and from Art. 62, = p cos p , and y p cos p . As f cos = cos(-Stf Wsin?, .'. v=/>sin p . \*f/ 9 3J 1/9 $ These equations are often written x = p cos 6, and y p sin 6. TRANSFORMA TION. 35 84. Schol. If the initial line is not the axis of x, we shall have (Art. 49), , and 3, = , sin * = /> cos The initial line being x 1 . 85. Exercise. Prove that the equations x sin x = p sin p , and y sin y = p sin ^ y y xx transform from oblique to polar co-ordinates. Proposition 19. 86. TJieorem.The equations transform from polar to rectangular co-ordinates. From Art. 83 x? = f) 2 cos 2 6, and y 2 = p z sin 2 . ' . adding, x* + y z = p 2 (sin 2 6 + cos 2 6) .'. by trig. x 2 + y 2 p 2 , and p = [/x 2 + y 2 cos = X also, COS 0= s t = 87. Example. Refer the point whose polar co-ordinates are \5, ^ j to rectangular axes with the origin at the pole, and the axis of x for the initial line. Ans. x = 4-33, y 2.5 88. Exercises. Prove from the figure given, that the equations, ^" p , f pf P x Po~T P x p A 36 THE POINT. transform from one system of polar co-ordinates to another, in which is the primitive pole, and 0' the new, p the old radius vector and />' the new, 00' X the initial line, and OO = p Q . The usual method, however, is to transform to rectangular axes, move the origin, and then transform to polar co-ordinates. Proposition 0.f 89. Theorem. The following are the equations of trans- formation, when the origin is moved to the point fa, y ), at the same time that the directions of the axes are changed. For by Art. 24 the equations x = x" X Q ", and y y" y /r will change the axes to a parallel position. On substituting these values of x and y, and then omitting the seconds, as they are not needed longer to distinguish the different systems of axes, we have, 1. Tlie equations (x - x ) sin x y = x f sin *' + y' sin *' (y y ) sin x = x ' sin X x + y' sin y x transform from oblique axes to oblique. 2. The equations x x = x f cos %' + y' cos &.' transform from rectangular to oblique axes. TRANSFORMATION. 37 3. The equations ... v ' x x Q = x + y cos transform from rectangular to oblique axes when x is parallel to x'. 4- The equations xx = x cos transform from rectangular to oblique axes when y is parallel to y f . 5. The equations (x - X Q ) sin x y = x' sin *' + y' cos * (y - y ) sin % = x f sin jjf + y' cos ^ transform from oblique axes to rectangular. 6. The equations (z-rgsm* = z'sin transform from oblique axes to rectangular when x is parallel to x'. 38 THE POINT. 7. The equations (y - y ) sin *= x' sin f + y' cos *' transform from/ oblique axes to rectangular when y is parallel to y f . 8. The equations x x = x' cos *' y' sin % y-y Q = x' sin % + y'coa% transform from rectangular axes to rectangular. 9. The equations transform from oblique to polar, co-ordinates. 10. The equations x-x =pcos p x transform from rectangular to polar co-ordinates. CHAPTER III. THE RIGHT LINE. Proposition 1. 90. Theorem. The equation y y\ 2/2 y\ represents a right line through two given points; in which x and y are the co-ordinates of any point of the line, and XL and y^ x 2 and y^ are those of the given points. For, let P be any point of the line through P l and P 2 , the given points. Then from similar triangles P : P.ft : : P& : P.& ^ >> \ a \ \ X This right line is conceived of, as indefinitely extended in either direction, and is called the locus of P. It may be drawn across any angle, first, second, third or fourth, according to the position of PI and PZ. mi ,. ,, T ,. 91. Schol. 1. Ine equation - - expresses the relation # # that must hold in order that some point (x 3 , y 3 ), (i. e., P 3 ), shall be upon this line. 39 40 ' EIGHT LINE. For evidently P 3 must coincide with some one of the infinite number of positions of P. Clear of fractions and we have, y l (x 2 # 3 ) + y 2 (x s x^) + y 3 (#1 #s) = 0, which, by Art. 31, is the relation which holds when a triangle reduces to a straight line. This is called the equation of condition .that three points shall be upon one right line. 92. The distances OA and OB are called intercepts, being the parts of the axes between the origin and the line. It will be convenient to use a and b to denote intercepts on x and y respectively. 93. Examples in either rectangular or oblique co-ordinates. (1.) What is the equation of the line through the points (3, 2) and (2, 4)? Ans.y = x 19. (2.) Through the points (2,S) and (4, 1)? Ans. 3y = 2x 5. (3.) Are the points (2, 5), (1, 1) and (1, 9) on the same straight line ? (4.) The points (3, 4), (1, 1) and (3, 5)? (5.) Write the equations of the lines through the points in ex. (3), and in ex. (4). (6.) Draw the lines whose equations are obtained in the examples of this article. 94. Exercise. Show that the form of the equation y y\ _ 3/2 y\ X Xi X 2 Xi is not changed by moving the origin to any point (a: , y ). Proposition 2. 95. Theorem. The equation -+H a b represents a right line ; in which x and y are the co-ordi- GENERAL EQUATION IN TERMS OF INTERCEPTS. 41 nates of any point of the line, and a and b are the inter- cepts. For, in Art. 90 let P l Ml on the axis of x, and P 2 on the axis of y. . . x l - OP l = a, and y^ x 2 0, and y 2 = OP 2 = b y0 b-0 .'. we have - x a a x _ +l=L / \ a b This equation is symmetrical, and of the zero degree when we consider both x and y and also the constants a and 6. In x and y only, it is of the first degree. 96. When a and b are given, we have the equation of a par- ticular line. But a and b may represent any intercepts, and in this sense the equation is said to be the general equation of a right line in terms of its intercepts. 97. Examples. Reduce the equations of the straight lines obtained in Art. 93 to the form a^b' i. e., so that the right hand member is +1: e.g., if 8y = x 7 then, transposing and dividing, x y~=i . ' . the intercepts are 7, and :z ^. Also show that when (Art. 90) Knv. = Ji= then a= ! ^Mi i and 6 = ,= 42 RIGHT LINE. Proposition 3. 98. Theorem. The equation = X 1 represents a right line through one given point ; in which x and y are the co-ordinates of any point of the line, sin Xi and T/! those of the given point, anci m = . sin For, from the triangle PQP l by trig. y y l : x x l : : sin QP^P : sin P-^PQ, . i yy^ n x = m, x x i sin y in which I denotes the direction of the line i. e., sin x is read, " sine of the angle between the axis of x and the line V 10 sin , 99. Cor. 1. When y r = <* = 0, - then m = tan- JC I JC ' cos x .'. in rectangulars m = the tangent of the angle which the line makes with the axes of x. 100. Cor. *.- we have, sin x COS x x l sn cos in which I = PP l is the distance of any point P from the given point P lt and is measured along the line. This equation may also be written EIGHT LINE THROUGH ONE GIVEN POINT. 43 101. Schol. 1. It is to be noticed that when the value of m is given in the equation a particular line is represented ; when it is not given, the equation may represent any one of the lines through P 1; and in this sense it is said to be the general equation of the line through one given point. 102. Schol. 2. In Art. 90 suppose that P 2 coincides with P lt then, - - 1 = m = some indeterminate quantity. This algebraic indetermination expresses the known fact that an infinite number of' different lines can be drawn through PI. Proposition 4. 103. TJieorem.The equation y b = mx represents a right line; in which x a?id y are the co- ordinates of any point of the line, b is the intercept on y sin^ and m= - sin ^ For, in Art. 98 let P l fall upon the axis of y. x l = and y l = b, y-b . . =m, or y b = mx. Similarly if P l be upon the axis of x, then x l a, and y l = 0. y y / - m. or x a = . x a m 44 RIGHT LINE. 104. Cor. 1. When2/ = 0, then m = tan l or ==cot* x m The equation y b = 7712: might therefore be written, y b = x tan ^ , or y = # tan * 4- ft. 105. Cor. . If 5 = 0, y = ma; is the equation of a right line through the origin i. e., if the equation contains no constant term, then the origin is on the line. If y = x, the line bisects the angle ^, and passes through the origin, and y = x + b is parallel to this bisecting line. 106. Cor. 3. When m = 0, then y = 5, and the line is parallel to the axes of x. When also 5 = 0, then y = 0, and the line coincides with the axis of #. When in the last equation of Art. 103, ~ = 0, then x = a, and the line is parallel to the axis of y. When also a = 0, then x = 0, and the line coincides with the axis of y. 107. Examples. (1.) Determine the intercepts of the right line through the point (1, 5), when m = - . Ans. a = 5-| , 5 = 4\ - (2.) Determine the intercepts of the right line through the point (4, <), when m = f . ^.ws. a = 7^ , b = -?^. (3.) Find the equation and intercepts of a right line through the point ( 2, 4), and perpendicular to the axis of y, when at = ^. -4ns. Equation is y = | 5. Intercepts, a = 10, b = 5. (4.) Find the values of m in the examples of Article 93. (5.) Show that the intercept a = . m 108. Exercise. Derive the equation of Art. 103 from the figure, and prove the equations of Arts. 90 and 98 from it, when at = 90. THE GENERAL EQUATION OF THE RIGHT LINE. 45 Proposition 5. 109. Theorem. Tlxe general equation of the first degree represents some right line; in which x and y are the co-ordinates of any point of the line, and A, B and C may each have any real value whatever. For if the origin be moved Fv \-^ to a parallel position by the equations of (Art. 23), viz. : x=x +x', and y=y Q + y', we obtain by substitution, , , p^ . . . (a.) \ Values for x and y have not yet been assigned, and we may evidently give them whatever values we please. Let them have such values that Ax Q + By Q + C=0. . . . (&.) There can be an infinite number of such values i. e., of positions of the new origin (X Q , y ), for if A, B and C are given, and we assign any value whatever to y , we can evidently find from eq. (6.) a corresponding value of x such as will verify the equation. Let the new origin (x Qt y ) be at any point 0', that satisfies eq. (6.) ; then eq. (a.) becomes Ax' + By' = 0. .......... (c.) Equation (c.) then represents the same thing referred to new axes that (a.) represented when referred to the primitive axes. If in (e.) x' = 0, then y'=0, which shows that this new origin is a point on the line, straight or curved, which is repre- sented by (a.) and (c.). We may omit the primes and write eq. (c.), Ax + By = 0, for the primes are used only to distinguish conveniently one system of axes from another. Now multiply by sin ^, or (Art. 13), Ax sin * - By sin y x = 0. 46 RIGHT LINE. Next change the direction of the axes. Substituting from Art. 69, ,-. A(x' sin x y +y' sin*') -B (x f sin % +y' sin ) =0. Rearranging the terms we obtain, (A sin *'-.5smf) x f + (A sin & -B sin %) y' = 0. . . (d.) Since the angles *', ^, ^, * , are not yet determined, and two of them, either *' and *', or *' and g', or * and *' or %' and *'> are independent, we may assign any values we please to the two, or affix such conditions as will determine their values. Let the two following conditions hold : 1st. A sin y' B sin *' = 0, and 2d. A sin *' - B sin |' < 0. Substituting the 1st condition in equation (d.), we have, by the 2d condition. From which we see that there is no point represented by this equation which does not coincide with the axis of x f , for y f = is evidently the equation of a line coinciding with the axis of x', but the axis of x f is a right line. . . (a.) re- presents a right line. 110. Cor. 1. To reduce the general form of the equation of a right line, Ax + By + C= 0, to the form of the equation in terms of the intercepts (Art. 95), trans- pose and divide by Q. , or , , , is the form ; in which _C_ A ' are the intercepts. ONE LINE IN TERMS OF TWO OTHERS. 47 111. Cor. 2. To reduce the general form to the form of one inter- cept (Art. 103), solve with reference to y, A C C A s * n x in which, as before, b = , and = y 112. Schol. 1. The equation represents two lines separately, for by the general theory of equations each factor = 0. 113. Schol. 2. If A l x + B\ y + Ci = and A. z x + < are the equations of the lines (1) and (2), then A^x^-\- B l y i -\- Ci = and A z Xi~\- -5 2 2/t + C z = hold respecting the co-ordinates of the point of intersection, (x h y t ). Eliminating, we have, C\ B>i C 2 -JDi , C/i AZ C 2 -A-\ 114. Schol. 3, The general equation of the first degree, viz., contains but two arbitrary constants. JV. B. We shall, for convenience, speak of " the line Ax+By+C= 0" meaning the line which the equation Ax + By + 0= represents. Proposition 6. 115. Theorem. The equation k, (A,x + By + Cl) +k 2 represents some right line passing through the intersection of the lines A l x + B l y+C l = and A& + By + d = ; in which Aa and h are any multipliers. 48 RIGHT LINE. For, by Art. 109 it is the equation of some right line, since it may be reduced to the form, (Afa + Ajc 2 ) x + (Bfa + BJc 2 ] y + Cfa + CJc 2 = 0, or, as it may be written, A 3 x + B$ + C 3 = 0. Moreover, the equation is evidently satisfied when both Ap + By + C^O, and A& + B#+C 2 = 0, provided the values of x and y are the same i. e., simultaneous, in the two expressions. But x and y can have the same values in lines (1) and. (2) only at their intersection ; therefore the line (3) passes through the intersection of lines (1) and (2). 116. Cor. 1. The equation of line (1) is, k (A& + By -f (?) = 0. The equation of line (2) is, & 2 (A. 2 x + By + (? 2 ) = 0, and that of line (3) is, h (A& + By + a) + 2 (A& + By + Q = 0. Add these together, and they vanish identically. Therefore, when the equations of three lines, after being each multiplied through by any constants & lf 2 and 3, can be added so as to vanish identically, the lines pass through one common point, and are called convergents, or a pencil of three rays. 117. Cor. 2. If lines (1), (2) and (3), whose equations are, intersect in a point, by elimination we obtain, , G (A A ~ A,B 2 ) + a (A 3 B, - A,B,) + a (A,B, - A,B,} = 0, which is the equation of condition that three lines intersect in a point. 118. ScJiol. When the constants of lines (1) and (2) have definite numerical values, then (1) and (2) are determinate lines. But in com- bining (1) and (2) to obtain (3) (Art. 115), it is still possible to assign at pleasure any values to ^ and # 2 , thus producing any one of an infinite number of lines (3), each of which passes through the inter- section of (1) and (2), and each of which satisfies the equation of con- dition in Art. 117. E. G., if line (1) is x + y + 2 = 0, and line (2) is x 2yl = 0, then line (3) is, 2x y + l = 0, when t : & 2 = 1, and is, 3x + 3 = 0, when ^ :k 2 = %, etc., etc. RELATION OF THREE ANGLES OF A TRIANGLE. 49 119. Exercise. Prove that the three lines (1), (2) and (3) enclose the triangle whose area is t, when [a (A& - A,S,) + a (A*B, - A jg.) + a (A A - A, (A A - 4A) (A A - A&) (A& - AA) CJA-C by subshtutmg *, = . V, = etc., in the formula of Art. 31. Also prove Art. 117 from the above. Proposition 7. 120. Tlieorem.The equation ^Y tan tan tan [ = tan Jj + tan 'j + tan *J ' */ expresses the relations which subsist between the angles? any plane triangle in which IQ. / x and / 2 are the direc tions of its sides. (Art. 12.) Y For (Art. 12), 360 = \ + \\ + ^ this being the sum of the external angles of the triangle. >_fc_Jl _1_ ^0 -! X) 2 as may also be seen from Art. 113. 123. Cor. 2. If = 00, then m = oo. . * . (e.) reduces to 1 + ^777-2 = #, or m x = -- , . . (Art. Ill) A,A 2 + B^ = 0. Either equation is the condition of perpendicularity of two lines. 124. Cor. 3.li l ^ = 1 then m, = 0, or (Art. Ill) 2 = 0', PERPENDICULARITY AND PARALLELISM. 51 .*. A 2 = 0, is the condition of parallelism to axis of x, i.e., the line B$ + 2 = is parallel to the axis of x. If b = 90, then w 2 =oo, or (Art, 111) = 00; x -B* .*. =0 is the condition of perpendicularity to the axis of x, i. e., the line A^c + (7 2 = is perpendicular to the axis of #. 125. Cor. 4.-Since tan =- = ll, (Art. Ill), . . . ft _ ' ^ "" m?) ^ and cos =-- 126. Examples. (1.) Given the equation to construct the three intersecting lines indicated by the equation, and find the co-ordinates of their point of intersection. (2.) Show numerically, by means of Art. 121 and the equations y = 3x + 5, 4y = x + 8 and y = x l, that the relation of Art. 120 holds. 127. Exercise. Prove that when the axes are oblique, Proposition 9. 128. Tlieorem. (Rectangular co-ordinates^) ^e line y = mtf + 6 2 is parallel to y = m^ + bi by Art. The line y y l =m 1 (x xj passes through (a*, 7/1), and is also parallel to y = mtf + 6 2 . 52 RIGHT LINE. The line A& + B$ + C 2 = is parallel to A& + B^y + Ci = by Art. 1%%. The line A l (x xj + Bi(y y^ = passes through (x lt yj, and is parallel to A& + B$ + Ci = 0. The line y = -- x + b- 2 is perpendicular to y = m& + b v by Art. 1%3. *"* _ 2 The line y y\ = - ( i) is perpendicular to y= m& m : 4- b and passes through (xi, 2/1). The same line is also perpendicular to y y l = m l (xx^) at the point (x 1? 7/1). The line B& A^y + C 2 = is perpendicular to A& + B^y C, = by Art. 123. The line B l (x x^ Ai(y 2/1) = is perpendicular to AiX + Biy + C! = 0, and passes through (xi, 2/1). It is also per- pendicular to A! (x Xi) + Bi (y 2/1) = 0, at the point fa, yj. 129. Example. Find the equations of two lines at right angles with each other, the first of which passes through the point (2, 5), and has tan |==#i while the second passes through the point (4, 1). Ans. = 2x^9 and 2 + x = 6. 130. Exercises. (Oblique co-ordinates^) Show that _ , x + i a , and y = m^x -f ^ are perpendicular. m l + cos y X (2.) (^cos^-^O^-CA is perpendicular to A& -f- B^y + ,-Q = 0. l j rrn l cos y COS * # passes through (#1, ?/i), and is perpendicular to y = m l x + &i- (4.) (vl, cos y - ,) (a- - *0 - (B, cos g - A,) (y - y,) = passes through (a; lt i/J, and is perpendicular to A& + B$ + C l = 0. ONE LINE AT A GIVEN ANGLE TO ANOTHER. 53 Proposition JO.f 131. TJteorem. (Rectangular co-ordinates.} The equations y = m 2 x + b 2 , and y yi = 'm 2 ( x ~ x z), which may by Art. 121 be written m l m Q m l m y=T , -- X + OB and y y=-^ -( x x \ 1 + ra^o 1 + m^ v represent lines making an angle with any given line y=m l x + b 1 , or y y l =m l (x arj, such that the tangent of this angle is m = tan ^. This is evident from Articles 103 and 121. It is also to be noticed that two lines can be drawn each making an angle of the same number of degrees with y = m r r + 6 1; but one makes a positive and the other a negative angle, correspond- ing to + m and m respectively, from which we have two values of ra 2 . 132. ScJiol. 1. If mi = 0, the given line is parallel to the axis of ar, . . y = zp m^c + & 2 and y y\ = T m (x x^) are the equations. 133. Schol. 2. If mi = 00, the given line is perpendicular to the axis of a:. Since in this case rax 1 -J- . . y= x + Z 2 , and y y l = (a? Xi) m m Q are the equations. 134. Schol. 3. If m } =m , then y = b. 2 and y ^ = are the equations. But if m^ = mo the equations become or = a:an 2 , and y y,= (a;- a;,) tan ^ ('|). 54 RIGHT LINE. 135. Schol. 4. If mi = , then x = and x x l are the equations. But if m l = , then the equations become y x cot 2 I ) + b. z and y y l (x x^) cot 2 I J. 136. Schol. 5. If m = 0, the line is parallel to the given line, and the equations become y = m^x + #2. and yy\'= r in^(x a^). 137. Sc/ioJ. tf.-If m = oo, then J^L^L = !^ = _ J_ t 1 + WiW ^ m + W! m . . y = x -\- b 2 , and y y^ = (x x^\ are the equations, m l m-i and the line is perpendicular to the given line. 138. Examples. (1.) Form the equation of a line making an angle of 30 with the line 7y x-^/3 + 2 = 0, and having an intercept on y of 4- Am. y= 7 -/S.x4, or y = j^ v /3, x -4. (2.) Form the equations of two lines making with each ether an angle of -5, the first passing through the two points (1, 2) and ( 4> #), and the second through the point (1, 3). Ans. y = x-\-l, and y = 3, or x = 1. Proposition 11. 139. Theorem. The equation x cos ^ + y cos -^ p = represents a right, line; in which p is the length of the perpendicular let fall from the origin upon the line, and P. and P are the angles between the co-ordinate axes and the perpendicular. EQUATION OF LINE IN TERMS OF DIRECTION COSINES. 55 Let ^ + 1 1 = represent AB (Art. 95). Multiply by p, f) f) *' a x + b By trig. - = cos ^, and -^ = cos Jf . * . x cos -^ + y cos P p 0. This is the equation of a right line in terms of the direction cosines of its perpendicular, and^> is always considered as positive. The equation may also be derived directly from the figure, Oe + cd = Oc. cos eOc + cP. cos dcP y cos y + x cos * =p. 140. Cfcr.J. If ? = 00, y P = ^ y =-P 90* x xxx X a; cos |! + y sin P p = 0, or, if a P, # cos a + y sin a > = 0. 141. Cor. . The value of p, from Art. 139, is^ = a cos ? = a sin J.- But by trig, we have from the triangle AOB sm; = ab sin 56 RIGHT LINE. 142. Cor. 3. To reduce the form Ax + By + 0=0 to the form x cos -^ + y sin -^ ^? = 0, let ^ = #0 in the formula of the preceding article; then p = -. By Art. 110, a = -7, 6 = -- y (or + o ) -a, .o c . ' . the form is x 4- y = Hence to perform the reduction in any case, divide through by I/'' (A' 2 + j5 2 ). It is to be noticed that if the reduction be applied to itself, the divisor is of the form j/(sin 2 a -f cos 2 a) = 1, which does not change the form. 143. Example. Form the equations, in terms of the perpen- dicular from the origin and its direction cosines, of the diagonals of a parallelogram, each of whose sides is 4, and one of whose angles is 60*, taking two adjacent sides as axes of reference Ans. ~ 2 xyll + ~^y y^ 8.464 = 0, or x + y = 4, and x cos 60 y cos 60 = 0, or x = y. 144. Exercise. Prove that with oblique axes the divisor corre- sponding to that in Art. 142 is Also, show that for the line y b = mx the divisor becomes !/(-? + 2m cos ID 4- m 2 ) sin to PERPENDICULAR FROM A POINT TO A LINE. 57 Proposition 12.-\ 145. Tfieorem.The equation (x l cos a + y l cos ft -p) =p l expresses the length of a perpendicular let fall from any point upon a given line; in which (#1, y^ is the point, p v is the length of the perpendicular, and x cos a + y cos /5 p = is the equation of the given line. For, if x cos a+ y cos ft p= is (Art. 139) the equation of / Pj.,^^ \ _ XL AB referred to 0, and we move the origin to P^ (Art. 23), the equation of AB then becomes #! cos 10) to meet the line x %y 3, and making an angle of 4$ with the axis of x, will be found to be TI = 88.05. \ A Proposition 13. 149. Tlieorem.The equation (x cos ^ + y cos & pj (x cos ^ 2 + y cos ^ 2 ^> 2 ) = is that of the line bisecting the angle between two given lines ; in which x cos ^ + y cos *y p l = 0, and x cos p ^ + y cos ^ 2 p 2 = are the given lines. For, by Art. 115 it is the equation of some right line as PP Z through the intersection of two others, as Mm and Nn. Take any point (x, y) on the line (P, in the figure) ; then (Art. 145) X COS and x cos cos = cos But by hypothesis the sum of the first members of these equations = ; . . p n p m = 0, or p n =p m . \ P is any point on the line of equal perpendiculars i. e., upon the bisector. . % (x cos p x l +y cos^ 1 -^) + (x cosf 2 + y cos^ 2 -p 2 ) =0 is the equation of the external bisector i. e., the bisector of the angle not containing the origin, as BP 2 . Similarly, (x cos % + y cos ^ jpj - (a cos ^ 2 + y cos ^ 2 -_p 2 ) = is the equation of the internal bisector AP 2 . INTERNAL AND EXTERNAL BISECTORS. 59 150. Cor. 1. The equation in rectangular co-ordinates of the bisector of the lines + C 2 = 0, is (Art. 142) By+C* 151. Cor. 2. The co-ordinates of the point of intersection of the lines x cos a v + y sin a } p l = 0, and x cos 2 + y sin 2 p 2 = 0, i sin a. 2 p 2 sin a t ??, cos a. t p 2 cos a v are (Art. 113), x i = . T^^ L -( \ and y t = : ; * r . sm (a! a 2 ) sin (! a 2 ) 152. Cor. 3 The lines x cos i + y sin c^ p l = 0, x cos 2 + y sin 2 ^? 2 = 0, x cos 3 + y sin a 3 j? 3 = 0, (Art. 117), intersect in a point, when ^?! sin ( 2 a a) +^2 sin ( 3 aj) +^3 sin (! a 2 ) = 0. 153. Cor. 4. By Art. 118 we have also, fa sin (a 2 a 8 ) +jp a sin (a 3 aQ +^ 3 sin (! 2 )] 2 - - - : - - -- - - - - ~T~ &(,. sin (a 2 3 ) sin ( 3 aj sin (i 2 ) x 154. Example. Form the equations in rectangular co-ordinates of the internal bisectors of the angles of a triangle, the co-ordinates of whose vertices are (1, 2), (3, 5), and (1, 4} I and show that they intersect in a common point. Ans. y = 0.312 x + 1.692, y = -1.55x 0.35, y = 19.39 x + 15.39 Co-ordinates of point of intersection. x 0.72, nearly, y = 1.466, nearly. 155. Exercise. Prove that in oblique co-ordinates =Q , cos is the equation of the bisectors of the lines C l = 0, and 60 EIGHT LINE. POLAR CO-ORDINATES. Proposition 14. 156. Theorem. The equation represents a right line; in which p is the radius vector of any point of the line, and p the length of the per- pendicular from the pole upon the line. For, by trigonometry, PCOB'=J,. But = cos ~ = ,. , , . This may also be written \ p cos (6 a) p, or jO cos a cos d -\- p sin a sin 6 =p. 157. Cor. J. If a = 0, then /> cos =p is the equation -of a line perpendicular to the initial line. 158. Cor. 2. The equation c represents a line through the origin making the angle c with the initial line. 159. Cor. 3. The angle between the lines p cos (0 a 1 )=p 1 and p cos (0 2 ) =p 2 is the same as that between^ and p 2 i. e., 160. Cor. 4. Two lines are perpendicular to each other if 2 -~ i = 90, and parallel if 2 a x = 0. 161. Cor. 5. The equation of the external bisector between the lines (1) and (2) is (Arts. 83 and 149), when a> = 90, p [cos (0 a,) + cos (0 - 2 )J =p z +p lt -n . . /m + n\ I'm n\ By trig. cos m-\- cos n = #cos I - 1 cos I - I. POLAR EQUATION OF RIGHT LINE. 61 Similarly for the internal bisector, cos 162. Schol. 1. The equation p cos (0 a)=p l cos (O l a) is the polar equation of a line through the point (p lt 0^, for, (Art. 156), each side of the equation = p. 163. Schol. 2. The equation (Arts. 83 and 91) p cos /?, cos #! /> sin p\ sin 0j /! cos X /f> 2 cos 2 /G! sin /> 2 sin 3 or, )0 [/>! sin (0 - X ) - p 2 sin (0 0,)] = Pl p 2 sin (^ 2 ) is the polar equation of a line through two given points (p lt 0^ and 0> 2 , ^,). Proposition IS. 164:. Theorem. The equation A cos # + .> sin 6 + C=0 represents a right line; in which p and are the polar co-ordinates of any point in the line. For, transforming to rectangulars, since, by Art, 83, x =/> cos 0, and y=/? sin 6, the equation becomes Ax + By + C = 0, which is the equation of a right line by Art. 109. 165. Schol. 1. Ap cos + JBp sin O+ C0 can be reduced to the form p cos (0 a) =p ; for by Art. 142, -A -B C -c S is of the form p (cos a cos + sin a sin 0) =/>. ^ . * . p cos (0 a) =p, when tan a = , and |? = A ' /' + 62 RIGHT LINE. 166. Schol. 2. The condition that three right lines given by their polar equations, pass through one point is found in Art. 152, and the area enclosed by three right lines in Arts. 54 and 153. 167. Exercises. Let the vertices of a triangle be (x lt yj, (x 2 , y 2 ), and (# 3 , ?/ 3 ). (1.) Find the equations of its sides (Art. 90). (2.) Show (Arts. 37 and 128) that the equations of the perpen- diculars which bisect its sides are \ (x, -x 2 )x + fa -y^y = \ (x? - * 2 2 ) + 1 0/i 2 - y!) (x 2 - xj x + fa-y 3 )y = j i Ctf - *3 2 ) + i W ~ 2/3 2 ) (x 3 -xjx + fa -y^y=\ (x? - xf) + \ fa 2 - y, 2 ). (3.) Show (Art. 116) that these three perpendiculars pass through one point, and find the co-ordinates of the point of intersection. (4.) Show that the equations of the lines through the vertices and perpendicular to the sides opposite them are (x l -x. 2 )x + fa - y 2 ) y = (x l - x 2 ) x z + fa - y 2 ) y 3 (x 2 x 3 ~) x + fa y 3 ) y = (x* a? 3 ) ^i + fa 2/s) y\ (x^ x^ x + fa yj y = (x 9 xj x 2 + fa yi) y*. (5.) Show that these three perpendiculars also pass through one point, and find its co-ordinates. (6.) Show (Art. 90) that the equations of the lines through the vertices and bisecting the sides opposite them are fa + y 3 %0 x (x 2 4- x s fa + yi %) a; (a? s + ^i - (7.) Show that these three bisecting lines pass through a single point, and that its co-ordinates are EXERCISES. 63 k (8.) Show that the equations of the three bisectors of the angles of a triangle are (x cos i + y sin a v p^ (x cos a 2 -f y sin a 2 p. 2 ) = (# cos 2 -f ?/ sin a 2 p. 2 ) (x cos a, + y sin , > 3 ) = (a; cos 3 + y sin a 3 p 3 ) (x cos i + y sin a : pi) = 0. (9.) Show that these bisectors intersect in a point, and find the co- ordinates of the point. (10.) Show how many of the points mentioned in (3), (5), (7) and (9) are upon the same straight line (Art. 91). (11.) Show (Arts. 146 and 148) that when a line cuts the sides of a triangle ABC (produced if necessary) in the points Jj, J/and JV, then AN L CM j NB ' LC ' MA ~~ y. B. Such a line as LN is a transversal. (12.) Show (Art. 31) that when lines be drawn from any point through the vertices of the triangle ABC, and meeting the opposite sides, EC, CA, and AB respectively in D, ^7 and F, then AF BD CE = FB' DO' EA N. B. The three lines meeting at a point are convergenis. (13.) Show (Art. 151) that the polar co-ordinates of the intersection of two right lines are _ P-* ~ 2 P&* cos ai ~ Pt ~ sin (! a 2 ) p l COS 2 p z COS ! .' sin a z 2 sin a/ CHAPTER IV. THE CIRCLE. 168. Tangent Line. If a secant line be passed through two points, P 2 and P 3 , of a curve, and the points be conceived to move continuously along the curve until they meet at P l (the secant continuing to pass through the points in all their positions), the secant in its limiting position is called a tangent to the curve at P lf and is said to touch it in two consecutive or coincident points at P r The equation of the secant or chord P 2 P 3 is (Art. 90), =J!l = ZL=yi m x - x* x 3 -x. 2 "When the value of the second member of this equation is determined from the equation of any particular curve, and substituted, and the points are then made consecutive, the equation becomes that of the tangent to the curve* The intercept on x is OA l = a^. The subtangent is a* XL The length of the tangent is , and its length is = ~\/y\ + (&i #i) 2 - 169. Normal Line. A normal line to a curve is so situated that it is perpendicular to the tangent at the point of contact. The equation of the normal can be obtained from that of the tangent by finding the equation of a line through the point of contact, and perpendicular to the tangent (Art. 128).** The intercept on x is * In Differential Calculus the general equation of the tangent line for all curves - = ' * * The general equation of the normal line is = . X 3/1 ^/l 64 GENERAL EQUATION OF THE CIRCLE. 65 OA 2 = <%] the subnormal is A 2 J), and its length is ^ a 2 . The length of the normal is PiA 2 = ]/yi a 4- (x l a.;) 8 . Proposition 1. 170. T7ieorem.The equation represents a circle; in which x and y are the rectangular co-ordinates of any point of the circumference, Xl and y l those of the centre, and r is the length of the radius. For, by Art. 26 in the equation r represents the distance between (x lt y,} and (x 2 , y 2 ). Let (x 2 , y 2 ) be every point at the distance r from the fixed point (x lt y^ ; then if x and y represent the gen- _ eral values of x 2 and y 2 , the equation becomes (x x l ) 2 + (y y l ) 2 = r 2 . But the point (x, y) is everywhere at the distance r from (# yj, and hence must be always in the circumference of a circle whose radius is r. The same equation can be proved directly from the figure. 171. Schol. 1. If the origin be at the centre then x l = 0, and y l = 0, If the origin be on the circumference, we have x* -f ?/, 2 = r 2 ; substitute . . x* -f y 2 2x& %,y = 0. If in addition, the axis of x pass through the centre, then y^ = 0, and . . # 2 + y 2 2x^x = ; or since now x l = r, by transposition, If the axis of y pass through the centre, x^ = 0. x 2 -f y 2 2y^y = 0; or since now y l =r, . x 2 = 2ry y 2 . These are all of the form x 2 + y 2 + Ax + By + (7= 0. 5 THE CIRCLE. 172. Schol. . Conversely , the equation x 2 + y 2 + Ax + By + C= 0, referred to rectangular axes, always represents a circle ; for it may be A written 4 which is of the form (x xtf + (y y^f = r 2 , A B in which x l = -- , and y l = -' If A* + 2 -4C>0, the circle has r = ~ \/ A* + B L -4C, and the point ( , ) for its centre. If A 1 + 2 4C= 0, the circle is the point (x lt yj. If A 2 + _S 2 JfC is the length of the . tangent from P to the circle . ' . Substituting, eq. (a.) becomes (x x^ 2 + (y 3/i) 2 = r 2 + p 2 . But, by elementary geometry, p 2 = PS l . PS 2 . /S^ and S 2 are in the same direction from P, and the rectangle . PS 2 is therefore positive. 2d. Let P be within the given circle. If PP l =d,t'he.u(x-x l ) 2 + (y-y l ) 2 = d 2 . . . (6.) is the equation of the circle through P with the centre P r If d 2 = r 2 p 2 , then p is the length of the tangent from P to its intersection with the circle (x x^ 2 + (y y^f = r 2 . .'. Substituting, eq. (6.) becomes (x xJ 2 + (y y l ) 2 = ^ p 2 . But, p 2 = PS l . PS 2 , by elementary geometry. S l and $j are in opposite directions from P; the rectangle of P&! and Pj is therefore negative. 176. Example. Find the rectangle of the segments of the secant, drawn from the point ( 2, 5), cutting the circle 4 ^ 2 -^3/ 2 = x - Am. 26. 68 THE CIRCLE. Proposition 177. Theorem. The equation represents a right line, called the axis radical of the two circles whose radii are r-^ and r 2 , and whose centres are at (xi, 2/1) and (x 2 , y^) respectively. For, expand and cancel, and we have, which (Art. 109) is the equation of a right line. If some point P i. e., (x, y) be taken on this line without both circles, then, by Art. 175, Y + (y - y,) 2 - n 2 = (x - * 2 ) 2 + (y - ytf - r* =/, which expresses the property of the axis radical, that the tangent drawn from any point of the line to one circle, is equal to the tangent drawn from the same point to the other circle. The line is real whether the circles intersect or not; and if it passes within one circle, it passes within the other at the same time. 178. Schol. 1. The equation to the line of centres is (Art. 90) The equation to the axis radical is x l . x+ C, hence these two lines are perpendicular to each other. 179. Schol. 2. The equations of the axes radical of three circles TANGENT LINE. 69 taken two and two whose centres are the points fa, y^), (x 2 , y 2 )> (3, 3/3), are (Art. 177) (x, -xjx + (y, - y,) y = } [fo* - xfi + (y x 2 - y 2 2 ) - ( n 2 - r 2 2 )] (*, - a,) a? + (y, - y,) y - f [(* 2 2 - *.') + (y 2 2 - y 3 2 ) - (r 2 2 - r 3 2 )] (0:3 - ar,) * + (y, - y,) y = j [(* 3 2 - xf) + (y 3 2 - ^) - (^ ~ rfi] which intersect in a single point called the (Art. 116) centre radical. Compare these equations with the equations of the parallels to them. Art. 167 (2). This may also Tbe more easily demonstrated as follows : if by the equation S\ = be understood (x a^) 2 -f- (y y^ 2 r* = 0, then will Si = 0, /S^ = 0, S* = 0, represent three circles, and the equations Si S 2 = 0, ^ ^ = 0, S 3 /Si = 0, will be those of their axes radical, which meet in a single point (Art. 116). 180.' Example. Find the axes radical of three circles, two by two, whose radii are respectively 2, 3 and ^, and whose centres are at the points (1, #), (5, 2} and ( 1, 3) ; and show that the three intersect in a common point. Ans. 8x 8y 19. y= 7. y = 13. Proposition 4. 181. TJieorem. The equation represents a line tangent to the circle x* + 2/ 2 = r 2 ; in which (2*1 2/0 is the point of tangency. For, if two points P 2 and P 3 be taken upon the circumference of the circle x 2 + y 2 = r 2 , the equations Co i t/9 I/ ctnci *t/o "i ?/o / cir\3 the equations of condition that P 2 and P 3 be so situated. The equation of the line P 2 P 3 is (Art. 90) y ~^- =^ 70 THE CIRCLE. But z 3 -* 2 2/3 + 2/2 Substituting this value in eq. (a.} we have as the equation of the line secant to the circle x z + y 2 = r 2 , through P 2 and P 3 . Let P 2 and P 3 be conceived to approach each other along the curve until they are consecutive points at P l (Art. 168). The secant line passing through these consecutive points will be the tangent, and we shall have, x 3 = x 2 = x 1} and y 3 = y. 2 = y r ij 11 x .'. eq. (6.) becomes r = -7, or Wi'yJf^-XBi + 'xf. ju a^ y l 182. Cor. 1. If the origin be changed (Art. 23), the equation becomes (x X Q ) fa x ) + (y y ) (y r y ) = r 2 , which is tangent to the circle (x # ) 2 + (y yo) 2 = ^ 2 , at the point fa, y^}. 183. Cor. 2. The intercepts of the tangent upon the axes are 184. Cor. 3. The subtangent T or, p 2 - %>! cos (6 - a) = r* - p?. . . (b.) This equation may also be obtained by transformation. EXERCISES. 196. Schol. The general form of the polar equation of the circle is p* + Ap cos + Bp sin -f C= 0, for, by trig, we may write eq. (6.) p 2 2ppi (cos cos a + sin sin a) (?* 2 /o/ 2 ) = 0, or, /o 2 (2pi cos a) /> cos (/>! sin a) p sin (r 2 p 2 ) = 0. Now place the constants, .2pi cos a~ A, 2pi sin a = J5, and (r 2 jO^) = (7; . * . /o 2 + ^4/> cos + Bp sin + C= 0. 197. Cor. If the pole be upon the circumference, PI = r, p 2 = 2pr cos ^., which has the two roots, p 0, and p = 2r cos ^. The latter is then the polar equation of the circle. If, however, the pole is at the centre, pi = 0, and /> 2 = r 2 , . . p = r is then the polar equation of the circle. 198. JSteeretees. (1.) Prove that the perpendicular "from the origin upon the j) tangent p= (2.) Show (Art. 172) that the equation X represents some circle through (& lt yi). Also find the equation when A or B is eliminated instead of 0. .y Show that the equation (x* + f) (xiy. t - x,yi} + C(yi - y,) x + (X 2 + represents some circle through the two points (x it ?/0 and (a^, y 2 ). Also when 5 and C are eliminated the same equation is (x* +y 2 ) (yi -y t ) + (^ +2A 2 ) (y, -y) + (^ 2 + yf) (y-yO - ^[[(ar x^ 2/ 2 + (a; 2 x) yi + (^i x^ y] = 0. 78 THE CIRCLE. (4.) Show that the equation (x* + y 2 ) [fe - *,) y 3 + (a;, - *,) y 2 + (ar f - a*) y,] ^ (#i 2 + yi 2 ) [(a? 2 a? 8 ) y + (a; # 2 ) y 3 + (a; 3 a;) yj + (^ 2 2 + y 2 2 ) [(a* ^) yi + (a?i a; 8 ) y + (a; ^) y s ] - fe 2 + y 3 2 ) [(a? arj) y 2 + (a? x) y l + fe # 2 )y] ^ represents a circle through the three points (a^, yO, (x 2 , y 2 ) and (z 3 , y 3 ). Also show that the equation of condition for four points to be upon the same circumference is expressed by substituting (x 4 , y 4 ) for (x, y) in this equation. (5.) Show (Art. 31) that the equation of condition of four points on one circumference (4) has this geometric significance : when /!, p 2 , p 3 , /? 4 are the distances of the points P lt P 2 , P 3 , P 4 on the same circumference, and P l P 2 P $ signifies the area of the triangle, Pi P 2 P 3 , etc. (6.) Show that x* + y 2 (! + 03) a; + G^OJ + By = represents the circle whose intercepts on the axis of x are Oi and a z . (7.) Show that represents the circle whose intercepts on the axis of y are bi and b 2 (8.) Show that represents the circle whose intercepts on the axes are a 1( a 2 , ^ and b 2 . (9.) Find the conditions that a circle touch each and both of the axes. CHAPTER V. EQUATIONS AND LOCI. 199. A plane locus is the line, straight or curved, described by a point moving on a plane according to some law. This law consists in the existence of & fixed relation between the x and y co-ordinates of the point, in all its successive posi- tions. An equation of a locus is used to express this law. 200. The connection between a locus and its equation is the fundamental idea of co-ordinate geometry. E. G. It was shown (Art. 109) that the locus of a point moving according to a law expressed by any equation of the first degree in x and y is some right line. Thus, in the equation Ax+By+C=0, let A=l, B=l and C=2, .'. x + y + 2 = 0, if any value whatever be assigned. to x, a corresponding value of y will be obtained from the equation ; and the points obtained by assigning successive values to x will be found to be all situated upon a fixed right line. 201. Variables. The co-ordinates x and y are called variables, or sometimes current or running co-ordinates. 202. Constants. The quantities A, B and C are called con- stants. When given, they cause the locus to be definite and fixed. If varied, they move the locus to some new position or effect some change of form. When unknown or undetermined, they are sometimes called variable parameters or arbitrary constants. Constants are co-efficients or exponents of variables. 79 80 EQUATIONS AND LOCI. 203. The equation of a locus is composed, then, of two kinds of quantities, variables and constants. Variables are the co-ordinates of a moving point, as (x, y), (ft, 6), etc., for general values ; (x l} yj, (p lt OJ, etc., for restricted values (Art. 19). Constants are the co-ordinates of a locus. JV. jB. We have found it convenient to use A, B, C, m, p, a, b, c, etc., for general values of constants, and A lt B l} Ci, m^ p^ etc., for restricted values. Also, x lt y 1} p lt O t , etc., are frequently considered to be constants. 204. Constants impose the restrictions to which variables are subject, and the conditions which a locus may fulfill. 205. A Condition is some fixed relation existing between two or more loci. An equation of condition is used to express this relation. E. G., It was shown in Art. 123 that mi = is the equation of w 2 condition , which is true whenever the lines y = m^x + b^ and y = m. 2 x + b. z are perpendicular to each other. 206. Transformation of Co-ordinates is the reference of any fixed or moving point to a new system of co-ordinates. 207. Equations of Transformation express the relation between the primitive co-ordinates and the new, and contain new vari- ables, primitive variables and constants. 208. Equations of numerical value are for the purposes of com- putation. When an equation of any kind becomes determinate, it is of this character, as, moreover, are the equations of ordinary algebra. 209. Cyclic symmetry exists in an expression when certain letters or suffixes are exchanged or permuted in a cycle (see Art. 34), to form its different terms or equations. DIMENSIONS. LOCUS. 81 E. G., In the equation of Art. 34 the suffixes are permuted in a cycle of three. In the equations of Arts. 167 and 198 a kind of double cyclic symmetry is noticeable. A symmetric arrangement is a useful aid to the memory, and greatly facilitates algebraic work. Proposition 1. 210. Theorem. Every term of any equation whatever rep- resents quantity of the same kind, be it volume, area, distance, mere number or other measurable thing. For the terms are connected by one of the signs + , , =; and since it is impossible to increase or diminish anything except by a quantity of the same nature, or to affirm the equality of unlike things, the proposition must be true. 211. ScJiol. A volume is said to be of three dimensions, an area of two, a distance of one, and a ratio, or mere number, of no dimensions. Space is at most of only three dimensions, but space of four, five or n dimensions may be used as a purely analytic conception. Proposition 2. 212. Theorem. Any single equation between x and y represents some plane locus. For it expresses some fixed relation of x and y, such that if x have a given value, y becomes known. By assuming successive consecutive values of x t and finding the corresponding consecutive values of y, we trace the motion of a point. A point so moving describes a locus (Art. 199). 213. Schol. The locus of an equation which has no terms except those which contain x and y passes through the origin. For let one of the successive values of a; be x = 0, then y = ; but these are the co-ordinates of the origin. E. G., In the equation if = 4px (see figure in Art. 243), if x = 0, then y~-=0. See also Arts. 109 and 171. It will hereafter be shown (Art. 458) that when in addition there are no terms of the first degree the locus passes twice through the origin ; 82 EQUATIONS AND LOCL and when there are no terms of the second or lower degrees, the locus passes three times through the origin, etc. 214:. Examples. Show the truth of the above proposition numeri- cally by constructing the loci represented by the following equations. x 3 x1 (2.) y = Proposition 3. 215. Theorem. Two simultaneous equations between x and y, one of the m th degree and the other of the n th degree, represent, in general, mn definite points upon a plane, which are the mn intersections of the loci of the two equa- tions, considered singly, as in Art. 212. For, the two equations represent, each singly, a locus. The values of x and y can be simultaneous that is, can refer to the same points, only at the points of intersection of the two loci. If the x and y of both equations are the same, the two equations will be sufficient to find by elimination the definite values of x and y, which are the co-ordinates of the points of intersection. But by algebra the degree of the equation resulting from elimi- nation between two equations, one of the m th and the other of the n th degree, is mn t and such an equation will have mn roots. There will therefore be mn definite values of x and y, and mn intersections of the two loci. 216. Schol. 1. To find the intercepts of any locus on the axis of x, let y = 0, which is the equation of the axis of x. If x = 0, we shall obtain the intercepts on y. 217. Schol. 2. Any number of these mn points may coincide, which will be indicated by a corresponding number of equal roots. Any number of the mn points may be at the origin, or at infinity, when the corresponding roots will be or oo. Any even number of the mn INTERSECTIONS OF LOCI. 83 points may be imaginary that is, impossible which will be indicated by corresponding pairs of imaginary roots. 218. Examples. Find the points of intersection of the loci repre- sented by the following equations. (1.) + = 1 Ans. (2.) y* = y? and y i = 4x. (0, 01 (0, 0), Ans. 2 Proposition 4. 219. Tfieorem. The sum or difference of two equations of loci, of the m* and n th degrees respectively, is, in general, an equation representing a locus passing through the mn points of intersection of the two loci. For, if S m = be understood to be an equation of the m th degree in x and y, and S n = one of the n th degree, and k any number, then S m kS n =0 is the equation of some locus, since it contains x and y; and since, when S n and S n = 0, we have S m kS n = also, the three equations are simultaneous at the points of intersection of S m = and S n = 0. 220. Examples. Construct the loci represented by the following equations, and show that they pass through the points of intersection of the loci represented by their component equations. 84 EQUATIONS AND LOCI. (1-) fr+y-Q + P + tf-Q^O. (2.) (x + y-l)-(x> + f-4)=0. (3.) 3(x + y-l)-(i? + y t -4)=0. (40 (x* + y' + 2x S) 2(x t + y* 8x + r)=0. Arts. 115 and 177 also furnish examples under this proposition. ^Proposition &.f 221. Theorem. Any equation of a locus, some of whose constants are general (that is, their values are unde- termined), can, in general, be made to fulfill as many con- ditions as there are independent, undetermined constants in the equation. For, a condition may be expressed by an equation containing some one or more undetermined constants, and two conditions by two or more such equations ; one equation suffices to determine one constant, or two equations two constants, etc. There can then be as many conditions that is, equations as there are un- determined constants. 222. Schol. 1. Any locus whose equation contains undetermined constants can, in general, be made to pass through as many given points as there are independent, undetermined constants in its equation. For, if the co-ordinates of one point, as (# lf y^, satisfy the equation, then, when x l and y l are substituted for x and y in the equation of the locus, the equation becomes the equation of condition that x l and y l shall be on the locus. Between this equation of condition and the equation of the locus, one constant can be eliminated. The same pro- cess for a second point would eliminate a second constant, and so on. / nt 223. E. G. Take the equation of a right line - + - = ./, . . . (a.) a b and subject it to the condition of passing through the points (x lt 3/1) and (# 2 , 2/ 2 ). If (x lt T/ : ) is on the line we must have CONSTANTS AND CONDITIONS. 85 and eliminating between equations (a.) and (6.), we obtain + (a- l )y = z _ a ay^ If (ar 2 , y*) is on the line, we must have ^+f =;,...*.) a 6 and eliminating between (a.) and (cf.), we find similarly ^(o-^^ ^.^ a ay 2 Now eliminate a between equations (c.) and (e.), and we have as in Art. 90. 224. ScJtol. 2. Any locus whose equation contains undetermined constants can in general be made to be tangent to as many given loci as there are independent, undetermined constants in its equation. For, the co-ordinates of the points of intersection of one given locus with this locus can be found by elimination, in terms of the constants of the two equations. If the condition be found that will cause two points of intersection to coincide that is, form the condition that we have equal roots the loci will be tangent to each other. This is evidently one equation of condition, and will fix one constant. A tangency with a second locus will fix a second constant, etc. 225. E. G. Take the circle (*-3r,)* + (y-yi)' = r, . . . (a.) containing three independent, undetermined constants, and cause it to touch the three right lines y = x-2, . . . (6.) = 2, . . . (c.) By eliminating bet\veen equations (a.} and (6.), we obtain EQUATIONS AND LOCI. and the condition that this quadratic equation in x shall have equal roots is, by algebra, whence we find which is the equation of condition that the circle touch (6.) Proceeding in a similar manner with (c.), we obtain From (a.) and (d.) we obtain Equations (/.) and ((/.} are the equations of condition that the circle shall touch (c.) and (d.) respectively. We have, then, three equations involving three undetermined quantities, # 1( y l and r ; hence by elimi- nation we can obtain such values as will cause the circle to touch all three lines (5.), (c.), (d.) at once. Subtracting (e.) from (/.), we obtain The roots of this equation are x l = 2 and y = 0. This value of Xi in (g.) gives r 2 ; and placing in equation (e.) xi = 2 and r = 2, we find Also if we make y^ = and x l =4rm equation (e.), we find Since a negative radius gives an imaginary circle, we can only use the upper sign of , or 0.828 + ; whence a?i = 4 (V ^) = 5 .172, or TANGENCIES OF LOCI, ETC. 87 Thus it appears that there are four circles which touch the three lines, as follows : O,l n, tf) ni O- / lV A Q . ** ^i *j yi r /V) ' 3d. ^-^-^v 7 ^, .Vi = 0, r = 2v'2 2; 4th. ^ = 6- 226. Examples* (1.) Find the equations of condition that the right line y mx + b may be tangent to the two circles Am. b = 2y'l + m\ and b = 6m (2.) Find the resulting values of b and m. Ans. b = 12 or T 12 and m - (3.) Show that the equations of the four lines which touch the two circles are yy-^5 = x + 12, and yy / ~ll = 5x 12, yy!$5 = x 12, and y j/77 = 5x + 12. Proposition 6'.t 227. TJieorem.The general equation of the n th degree- is. e., an equation in which all the constants have general values can be made to fulfill | n(n + 3} conditions. For, the general equation of the first degree contains two con- stants independent of each other, that of the second degree 2 + 3, and that of the n ih degree 2+3 + 4+ ..... + (n + 1], or, by summing the series, ~ n(n + 3} constants; and hence (Art. 221) can fulfill I n(n + 3) conditions. EQUATIONS AND LOCI. 228. Scttol. Some curve of the second degree can be made to pass through 'smy five points, or be tangent to any five given loci (Arts. 222 and 224). Proposition 7. 229. Theorem. Transformation of co-ordinates cannot change the form of the locus, nor the degree of any equa- tion transformed. The/orm of the locus is unchanged, for that depends upon the relation of the different positions of the moving point to each other, and not upon the co-ordinates by which relation is expressed. The degree is unchanged, for in any transformation by equa- tions of the form x = m l x' + m 2 y' + m 3 y = n l x' + n 2 y' + n 3 (Art. 89) the values of the new variables being of the first degree in terms of the primitive, x 2 , or xy, or y 2 could at most be com- posed only of terms containing x' 2 , x' y' ', x' 2 , x f , y f , etc. ; there- fore the degree of the expression in the new variables could not be increased. Neither could it be diminished; for if any transformation could cause the resulting equation to be of lower degree, then, by a retransformation to the original expression, an equation of higher degree would be obtained, which has been proved to be impossible. 230. Example. Kefer the locus represented by the equation a: 2 + y* = 9, in rectangulars, to new axes in which y x = 60 , making the co-ordinates of the centre (3, 4), thus showing that the degree of the equation is not changed by the transformation. Am. x' 2 + 2/ 2 + of y' + 6x' The equation x* + y* = 9 represents a circle, and the new equation is evidently also that of a circle, since it is of the form of Art. 174. PROJECTIONS OF LOCI. \ /*> /' Proposition 8.f O, ' 231. Theorem. When, in any transformation of co-brcU- nates, some of the constants are undetermined, the new axes may, in general, be made to fulfill as many con- ditions as there are independent, undetermined constants in the equations of transformation. For, evidently, definite values must in some way be assigned to the undetermined constants, in order to fix the position of the new axes. One equation of condition is, in effect, the determi- nation of the value of one constant, two conditions of two con- stants, etc. (compare Art. 221). 232. Schol. There are evidently four independent constants in the most general transformation possible; for the axis of x' can be made to pass through any two points, and i/ through any other two ; or each can be made tangent to two different loci, etc., etc. Proposition #.f 233. Theorem. Replace x by -x', or y by ~y', in any a o equation of a locus, and the resulting locus will be an orthogonal projection (Art. 60) of the first locus upon some oblique plane. For, evidently, if the axis of x be supposed to pass through 0, and to be perpendicular to the paper, and the angle YOY' be such that -=sec YOY 1 '; then, * if the locus of (x, y) is in the plane XOY, when y=^y', or , = % the locus of (x, y'}, in the plane XOY', will be at the foot of the perpendicular upon XOY' through (x, y). The same may be proved if x = -x r . 90 EQUATIONS AND LOCI. 234. Schol. When p is replaced by in any equation expressed in polar co-ordinates, the scale of curve represented is thereby changed ; e. g., if m = 2 the scale is doubled, and if m = - the scale is decreased one half. /i When is replaced by , the loops or other parts of the curve which were contained in any angle t are caused to be contained in n times that angle ; e.g., if n = f , the part of the figure which was described by any single rotation of the radius vector through 360 will be com- pletely described during the rotation of the radius vector through 180. This will appear more clearly in the chapter on polar curves and spirals. Proposition 10. 235. Tlieorem. The general equation of the n th degree represents not only the curves peculiar to that degree, but also all curves represented by equations of inferior degrees. For, since in a general equation the constants may have any finite value, among other values are those which will enable us to separate the equation into factors, whose product is of the n th degree, and equal to zero. Since the product = 0, by the " Gene- ral Theory of Equations," each factor = 0, and so represents a curve of lower degree. 236. Schol. The product of two equations of loci represents at once the two loci. E. G. Take the two equations x + y and x y = 0, whose product is x 2 y 1 0. . * . y = dr every value of x we have two values of y, equal, but with opposite signs. It may also be noticed that the product of two equations of loci put equal to some constants represents a locus which approximates some- AXES OF SYMMETRY. 91 what to the two loci. -27. G. The equation x i y l = l represents a curve approximating in position to the lines x + y = 0, x y = *. e., x^-y l = (see fig. in Art. 358). Proposition 11.^ 237. Theorem. /, the conic is called an hyperbola. 93 94 THE PARABOLA. 241. Exercise. The equation is the general form of /> ed in rectangular. 242. A Focal Chord, or Parameter, is any chord of the curve passing through the focus. The Latus Rectum, or parameter of the principal axis, is parallel to the directrix. A Diameter is the right line which is the locus of the centres of parallel chords, as will be shown hereafter. The Centre is the point of intersection of diameters, as will be shown hereafter. The Vertex is at A, in the figure of Art. 239. Conjugate Diameters are so situated that each bisects a system of chords parallel to the other. . Supplementary Chords extend from any point of a curve to the extremities of the same diameter. THE PARABOLA. Proposition 1. 243. Tlieorem.The equation represents a parabola; in which x and y are the rect- angular co-ordinates of any point of the curve, and 4p is the latus rectum, when the origin is at the vertex, and the axis of x is the axis of the curve. RECTANGULAR EQUATION. 95 For, if (Art. 240) e = l, then y* + (x2p)* = a?. Beducing, y 2 = 4p(x-p). ..(a.) Let P coincide with A ; then y = #. .'. OA=x=p. But (Art. 239) if e = l, then f> = S. .' . OA=AF-, . ' . the vertex of a parabola' bisects the distance between the directrix and focus. Move the origin to A, then x =p, and y Q = 0', . ' . (Art. 23) x = p + x', and y = y'' Substituting these values of x and y in (a.), we have, y' 2 = 4px', or y' = g Omit the primes : . * . y^ 244. Cor. 1. If the origin be moved any distance x = a on the axis of x, the equation becomes (Art. 23) y 2 = 245. Cor. . The equation y 2 = ^[pa: represents a parabola in the position AOB, with the axis of y tangent at the vertex. y* = 4px represents A^OB^ with axis of y tangent at the vertex. x* = 4py represents A^OB^ with the axis of x tangent at the vertex. rr 2 = 4py represents A 3 OI> 3 , with the axis of x tangent at the vertex. 96 THE PARABOLA. 246. Cor. 3. If in the equation y 2 = 4px we let x=p, then y = :2p = j- (latus rectum'} : and (Art. 243) the distance from the vertex to the focus, or from the directrix to the vertex, is p = j- (latus rectum). Also, if y 2 = 4p%, then x : y : : y : 4p t ' i n the parabola the latus rectum is a third proportional to the abscissa and ordinate. If (#!, 3/1) and (x 2 , y 2 ) are upon the parabola, then y* = and 2 243. Schol. 1. Since p = d = x' +p, omit the prime, and the equa- tion p = z -\-p is called the linear* equation of the parabola. By it the parabola may be constructed. 248. Schol. 2.T he equation p = d enables us to construct the parabola by continuous motion, with the aid of a ruler CD, a triangle ABC, and a thread equal to AB fastened at A and F. If a pencil P be kept in contact with the triangle along AB, and the triangle slide along the ruler, evidently PB = PF, and the locus of P is a parabola. A Proposition 2. 249. Tlieorem.The equation represents a line tangent to the parabola y* = 4px point (xi, ?/i) of the curve. * An equation of the first degree ia frequently called a linear cquati TANGENT. 97 For (Art. 90), **-M /j> \ represents a right line through two points P 2 and P 3 . If these points are upon the parabola, the equations of con- dition which must hold are y 3 =4p x 3) and y} Subtracting, + 2/3) = 4p fe - 3 4 . ". substituting in (6.) - -= _f . . . . ( c ) x # 3 2/ 2 -t- 2/3 If P 2 and P 3 coincide at P D then y 2 = y^^=y lt and (c.) becomes, rf>~ which is the equation of the tangent line. From this w/i - y? = %P X ~ P X I '> but 2/i 2 = 4pzi, * yyi= -P ( x 250. Cor 1. At the point of contact the equation becomes y\=4pxi, as it should, since the point P l is on the parabola and on line also. 251. Cor. 2. If y 0, then 2p (x + Xl ) = 0, . . x = - x,, .'. the intercept a = x l ; i.e., AO=OAi, and AAi, which is called the subtangent, is bisected at the vertex. Also (from ^m. tri's) AO : OA l : : AB : P lt . ' . AB = BP V . 98 THE PARABOLA. 252. Cor. 3. If y = 0, then, a = x. .'. (Art. 247), Pl =p + Xi =p + a. i.e., AF=FP l , . . AFPi is an isosceles triangle, and FAP l = AP^F. But if ED is parallel to AX, then FAP, = EP^A. . . the tangent A T bisects EP^F, and EAFP^ is a rhombus whose diagonals bisect each other at right angles on the axis of y. Again, from the figure, DP l T= EP^A. . ' . AP,F=- DP T. If CP l is the normal at P lt then AP^= CP 1 T=90. Subtract .-. AP.C- AP,F= CP,T- DP^T. Also (from sim. tri's) AB : BP, : : AF : FC, . ' . (Art. 251) AF= FO = Pl . . ' . CFPi is an isosceles triangle, and FP 1 C=P 1 CF. 253. Cor. 4 If x = 0, then b = = y = yi But y l = 2^/px^ . . b = y'px lt and AB = y'aT+V 1 = yV +_pa^= v /^ l (Art. 247). Again a : b : : yV + V 2 : r, (= FB) or a?! : -\/px : : i/p^ : r. .' . r Also from (sim. tri's) it may be shown that and that CP l = 2BF= 254. Cor. 5. From the figure %p i = tan = tan r. y\ x T, , . tan 2 r By trig. sin 2 r - - , J TANGENT. 99 ^^TTV^ y? But = * .'. sin 2 r = But sm 2 r a form of the linear equation of the parabola of some importance ; in which p l is the radius vector of P,, and r the angle between the tan- gent at PI and the axis of x. Also since -- = tan r ; sin T = - ; . ' . 2 p cos r y, sin r = 0. cos r 255. Examples. Construct the tangents represented by the follow- ing equations, with the parabolas to which they are tangent, finding the equations of the parabolas, when y l 4- (1.) y (2.) y (3.) Find the equation of the tangent to the parabola y* = 1 when ^ = 9. Ans. 3y = 2x + 18. Proposition 5.f 256. T1ieorem.The equation P y ~ mx + also represents a line tangent to the parabola y* 4p%\ v which m = = tan . y\ 100 THE PARABOLA. i) and y? = For, if (Art. 249) yy l = then, by substitution, yy l = ' y = This is called the magic equation of the tangent line. , or #! = , 4p y = m# + m 257. Schol. 1. The locus of the foot of the perpendicular from the focus upon the tangent line is the tangent at the vertex. For, the equation of the tangent may be written my m' 2 x =p, and the per- pendicular to it through the focus, (p, 0), is (Art. 128) my + x p. Subtract, . . (1 + m 2 ) x = . . x = 0, which is the equation of the tangent at the vertex. This enables us to construct a pa- rabola approximately by drawing successive tangents, as in the figure. 258. Schol. 2. The locus of the intersections of tangents perpen- dicular to each other is the directrix. P. For, = mz -f 1 represents some tangent ; in which equation let us replace m by - (Art, 123), y = mp represents a second tangent per- m pendicular to the first. Subtract, etc., equation of the directrix. x = p, which is the 259. Exercise. Prove that the equation y = m<& + is the locus ra of the intersection of the tangent with a line drawn from the focus, and making with the tangent line an angle whose tangent = m . POLE AND POLAR. 101 Proposition 4. 260. Theorem. The equation also represents a right line which is the chord of contact of two tangents drawn to the parabola y. 2 = 4px from the external point (ap lf y^. For, let P 2 P 3 be the chord of contact of the tangents P t P 2 and P,P S . ^ ^ . . . (d.) represents P 2 P 3 . JO JC<* %Cn fcCo From Art. 90, From Art. 249, yy 2 = 2p (x + x 2 ) .. (e.) represents the tangent at P 2 , and if it passes through P v the co-ordinates of PI must satisfy (e.). = 2p (x l + x 3 ) . . . for.) - = , substitute in (d.). . . similarly subtract .-. clear of fractions. . * . yy l y{y z = %px Add (0r.). . * . yy l = 2p (x + x^ represents P 2 P 3 . PI is called the pole, and P 2 P 3 its polar, with respect to the parabola. 261. Exercise. If the pole is on the directrix, show that the polar is a focal chord. Proposition 5. 262. Theorem The equation also represents a right line which is the locus of the inter- sections of the pairs of tangents to the parabola ?/ = 4px, ' drawn from the extremities of all chords which pass through any fixed point (x it yi). 102 THE PARABOLA. For, let P! be the fixed point through which all the chords pass, and let QiQ 2 be any chord through P r If the tangents at Q l and Q 2 meet in some point P 2 , then (Art. 260) yy z = 2p ( is the equation of the chord 1%; an( ^ at -^i this equation becomes y\yz = ^P ( x \ + #2) ('*) Similarly, if the tangents at the extremity of anothejr chord through P l intersect at P 3 , y^ = gp (x l + x 3 ) . . . (fa) But (Art. 90), , --~^f - - (<) represents P 2 P 3 . x-x Eliminate as in Art. 260. . * . yy l =%p(x + xj represents P 2 P 3 . Pi is called the pole, and P 2 Ps its polar, with respect to the parabola. 263. ScJtoL The tangent (Art. 249) yyi = 2p(x + x l ) is the par- ticular case in which the pole is upon the curve, and consequently upon its own polar. 264. Examples. (1.) Given a parabola whose latus rectum is 8, to find the polar of the point (#, 7). Ans. 7y = 4x+ 12. (2.) The latus rectum of a parabola is 4 1 find the pole of the line (3.) Given a parabola cepts of its polar. z, and a point ( 4, 10), to find the inter- Ans. a = 4, b = ~. NORMAL LINE. SUBNORMAL. 103 265. Exercise. If the focus is the pole, the directrix is the polar. Proposition 6. 266. Theorem. The equation y-yi = -;z(*-*i) vp is that of a line normal to the parabola y i ~4px at the point (x l} ?/i). For, resuming the equation of the tangent line (Art. 249), J jl . \ I/) yi we have (Art. 123), y~y\ = ~~^~ (% #i) perpendicular to the tangent, and it passes through the point of contact (x l} y x ) by Art. 98. 267. Schol. 1. The equation y y l ^- (# # ) also repre- 2p sents the normal when the origin is at any point on the axis of x. Move the origin so that the axis of y shall pass through the point ?/ oc (#i> y\) i. e.,letxi = 0; then _ -f = 1, and (figure of Art. 249) y\ %p AiC=%p . ' . subnormal = a constant. 268. Schol. .f Let m= - y - = tan ^, . . y l = 2pm. Also, since y\ = 4px\, " Xi=pl \=pm z . Substitute these values in y y l = (x #j), then, y + 2pm = m(x prri?) or, y = mx pm (2 + w 2 ), which is the magic equation of the normal to the parabola. 104 THE PARABOLA. 269. Examples. (1.) Find the equation of the normal to the parab- ola, when p = 2 and x = 32. Ans. y = J^x -f 144- (2.) Find the equation of the normal to the same parabola, passing through the point (10, 4) not on the curve, the equation of condition that the normal shall pass through a point (x. 2 , y 2 ) being (Art. 266) y\ y\ Am. x l = 24 is normal at (8, 8). Proposition 7. 270. Theorem. The equation represents a\ parabola inferred to a tangent, and a parallel to the axis through the point of contact; in ivhich p' is the distance from the focus to the point of contact, and 4p f is the length of the focal chord parallel to the tangent. For, transfer the origin from the vertex to some point O f upon the parabola y 2 = 4px. . . . (I.) The equa- tions of transformation are (Art. 89, 3), i Mw all /F x = x l + x f + y' cos T when we make the axis of x f parallel to the axis of x, and the axis of y' the tangent at 0'. Therefore, substituting these in (I.), (y l + y f sin r) 2 = 4p (x 1 + x r +y r cos r), or expanding and arranging, ' (#p cos r y l sin r). y' 2 sin 2 TANGENT AND DIAMETER AS AXES. 105 Bat, since (x lt y t ) is on the curve, y? J t ,px l = 0, and (Art. 254), %p cos r y l sin r = 0, hence y' 2 = -^- *' ; . . (Art. 254), y' 2 = ^ x'. bill 4 Again (Art. 252), f> l =AF= O'C. Let x '= Pl = 0'C. .'. y' 2 = 4(> l 2 . .'. y' = 2 Pl = CQ. If, for convenience, p l =p', then, 4pi = 4p f is the- length of the focal chord parallel to the axis of y'. Hence, omitting the primes, y 2 = 4p f x. 271. Cor. Since y= #i/f&, every positive value of x gives two equal values of y, or the axis of x is the bisector of all chords parallel to the tangent, and is therefore a diameter, and all lines parallel to the principal axis are diameters. 272. Schol. Since parallel chords all pass through the same point (oo!, oo 2 ) at infinity, and since for this point x\ = oc^ and y^ = oo 2 , the equation of the polar of this point (Art. 262) yy^ = 2p (x + x t ) v , i .. or v= -- r becomes y , or y = a constant. I his polar y\ 2/1 3/1 of parallel chords is then (Art. 106) parallel to x, and cuts parabola at the point of contact of the tangent parallel to the chords. Hence the locus of the intersections of pairs of tangents at the extremities of parallel chords is the diameter bisecting those chords. Proposition 8. 273. Theorem. The equation represents a parabola; in which p is the radius vector, and the variable angle of any point of the curve, mea- sured from the vertex, when the pole is at the focus, and l = 2 is the semi-latus rectum. 106 THE PARABOLA. For, y 2 =4p(%+p) is ^ e equation of a parabola referred to the rectangular axes through the focus. Transform by the equations x = p cos 6 and y = p sin 6. Cancel V l-cos6) &p(l cos 0) ~7in 2 l-cos?d ' 2p , p P = TT 71i> and ~P = T^ 1 + cos d' 1 cos 6 which are the positive and negative values of p corresponding to the angle 6. 274. SchoLBy trig, /? /J 1 + cos = 2 cos 2 -, and 1 cos = 2 sin 2 -. 2 2 COS0 0' COS 2 __ 275. Exercise. Prove that cos is the polar equation of the parabola when the pole is at the vertex. Proposition 9.f 276. Theorem. The equation represents a line tangent to the parabola p = at 1 + cos the point (j> lt 0J, the pole being at the focus. POLAR EQUATION OF TANGENT LINE. 107 For, yy l = p(x+x l + 2p) is the equation of the tangent line referred to rectangular axes through the focus. Transform by equations x = j0cos#, and y=psmO', . . pp l sin 6 sin O l = 2pp cos 6 p p l cos l + 4p 2 . 'l + COaOj? Substitute, clear of fractions, transpose and solve for p, r cos (6 - 0J + cos 6 cos (6 - L ) + cos 6' 277. ScJiol. 1. If = 1} this becomes the equation of the curve, as it should. 278. Schol. 2. By trigonometry; I P cos 2 (d 0!) + cos 279. Examples. (1.) Construct the tangent to the parabola p = 1 at the point whose vectorial angle is O l = 60, and find 7 + COS0 the angle which it makes with the initial line. .Ans. The angle = 60. (2.) Find from (Art. 280) the normal to the same parabola at the same point. 280. Exercise. Prove that the equation of the normal, when the pole is at the focus, is *4 If = l} this also becomes the equation of the curve. CHAPTER VII. HYPERBOLA AND ELLIPSE. Proposition 1. 281. Theorem. The equation 7>2 represents an hyperbola with the origin at the centre; in which a 2 and b* are the squares of the intercepts on x and y respectively i. e., the semi-axes. For, y x8p)* = #tf. . . (a.) is by definition (Art. 240) OA the equation of an hyperbola when e > 1, with the origin at 0. If y = 0, then x 2p = d- ex, and . * . is the intercept OA 2 . 108 = Sp_ 1 + e intercept OA l} and X Q = 73 JL C/ RECTANGULAR EQUATION OF THE HYPERBOLA. 109 Let CA^ = + a, and CA 2 = a, then (Art, 9), CA t = J- (OA t - 0^ 2 ) = - = - = a. Again, 07=1 Move origin to C (Art. 23), . . y=y', and x --=x r --- , 6 and substituting these in (a.), ... y '2 = a 2 - x' 2 - e 2 (a 2 - x' 2 ) =(1- e 2 } (a 2 - x' 2 ). If x' = 0, then y /2 = a 2 (l- e 2 ), the second member of which is negative when e>l, .'. 'let a* (1 #)=-&; then, T/' 2 =^j r (a 2 - x' 2 }, or ~ + ^ = 1, . . . (b.) (dropping the primes) represents the same curve that (a.) do^s. 282. Cor. 1. This curve is in form symmetrical about both the axes of x and y, for every value of x gives two values of y numerically equal, but of opposite signs, and vice versa. There are no real values of y in this curve between x + a and .>; = a, as may be seen from the equation, when put in the form, 110 HYPERBOLA AND ELLIPSE. 07.2 283. Cor. 2. The latus rectum ED = - a For, if e e let x f = ae ; .-. y ; = ^ (a *- aV ) = 4. .-. Ky t (Id 284. Cor. 3. Since a 1 (1 - <*7 = - S ! , or a* + V = oV, for, (Art. 283) CF = ae = i/a? +b\ and from the fig. A& = j/a 2 + b\ A 1 / & C? Also, C0 = - . 285. Schol. 1. The axis A^ = 2a is called the transverse axis. The axis B^B 2 = 2b at right angles to the transverse axis is the conju- gate axis. 286. Schol. 2. The hyperbola becomes equilateral or rectangular when the axes are of equal magnitude ; X* y 2 = a 2 is the equation of an equilateral hyperbola. 287. Examples. Find the value of CF lt CA l} CB^ and ED for the hyperbola represented by each of the following equations, and con- struct their directrices, vertices, foci and the extremities of each latus rectum. (1.) L-| %- = !. Ans.a=3, b = %/ 1. 9 4 (2.) + _ =: 1, Am. a = l,b = S^l. 9 9 (3.) ^-^ = jr - Am.a=2^b = Vl. (4.) Show that the latus rectum is a third proportional to the axes 2a and 2b. (5.) Show that a is mean proportional between CF l and CO. RECTANGULAR EQUATION OF THE ELLIPSE. Ill Proposition 2. 288. Theorem. The equation represents an ellipse with the origin at the centre; in which d* and 6* are the squares of the semi-axes, and a > 6. The demonstration of this proposition is identical with the preceding, with this ex- ception, that since (Art. 240) e < 1. The lettering of Art. 281 will be found to apply to this figure, and the results will therefore agree with those in Art. 281 on changing the sign of b 2 . x 2 v 2 289. Cor. 1. The curve is symmetrical ahout the axes of x and y. The values of y are all real between x = + a, and x = a, but I imaginary beyond, since y = -\/a 2 x'\ #7,2 290. Cor. 2. The latus rectum = (Art, 283). 291. Cor. 3 Since a 2 (1 - e 2 ) = b\ . . e = l C=ae = i/tf^F, and 292. ScftoZ. Jf. The axis ^^ = ^0 is the transverse or major axis, and ^ J5 2 = #& is the conjugate or minor axis. 112 HYPERBOLA AND ELLIPSE. 293. Schol. 2. The ellipse becomes a circle when a 2 = b 2 = r 2 . Then e = 0, and the equation becomes x 2 -f y 2 = r 2 . 294. Schol. 3. When 5 > a, then e 1 = V /al " y . i s imaginary, but CL e 2 = - is real, . . b is the major and a is the minor axis. 6 295. Schol. 4. The equations of the ellipse and hyperbola may be written together, in which the sign + is for the ellipse and for the hyperbola. 296. Schol. 5. The equation 2ii _j_ 0/ .Vi) = i represents a 2 b' 2 an ellipse or hyperbola whose axes of reference are parallel to its axes of figure ; in which x^ and y l are the co-ordinates of the centre. This may be shown by moving the origin (Art. 23) in the equation x 2 i/ 2 -| '-^^ = 1. If Xi = zh , then the origin is at A l or A 2 , and we /y.2 Ortnr- njl have for the equation at the vertex -- - --- h -~3 = ^' ~~ , or y z = x^ x\ 297. Schol. 6. The equation of the ellipse is (Art. 289), -y t = i/a 2 x 2 , and that of the circle (Art. 171), y c = i/a 2 x*. b For any value of x common to the ellipse and the circle we have, . . y c = - y e , or = -. . ' . by Art. 233 the ellipse is a pro- & V* jection of the circle when the major axis is the line common to both. The same may be proved of the circle on its minor axis i. e., for any x c b value of ?/ common to both, = . x* a CONJUGATE HYPERBOLA. 113 298. Examples* (1.) Change the sign of the term containing y 1 in each of the equations of Art. 287 from to -f , and find the position of the foci, etc., of the ellipse represented by each equation. (2.) Show that the proportions stated in Art. 287, (4) and (5), are true in case of the ellipse. Proposition 3. 299. Theorem. The equation -a represents an hyperbola conjugate to a? y 2 -4- -- / a 2 -6 2 ~ i. e., the transverse axis of the one is the conjugate axis of the other, and vice versa. For, evidently this represents the curve whose vertices are at B^ and JB 2 (Art. 281), and it may be considered to be derived from x 2 y 2 -- h = / a 2 * -b 2 x 11 by the exchange of - and 7. Also since a 2 is the square C6 of a semi-axis, that axis is imaginary, while the other is real. Similar corollaries and scholia apply to this proposition and Proposition 1. 300. Cor. In the conjugate hyperbola for, CF t = A& = CF l = i/a + b\ and CB l = b. Also, the distance from to the directrix = - = 2 s 114 HYPERBOLA AND ELLIPSE. Proposition 4* 301. TJieorem.The equation : - a? y 2 +-Z=1 a 2 b 2 represents an imaginary ellipse. For, from this equation y = - y a? x 2 , hence all values of x give imaginary values of y ; still either e l or e 2 is real, and has the same value as in the real ellipse (Art. 294). v -Proposition . 302. Theorem. The equations ^+ y --1 nd - ~^ 1 a 2 ^b 2 a 2 d 2 ~ i ~b 2 d 2 ~ represent conies having the same foci, and are called con- focal conies. For, in the first curve (Art. 288) a 2 e 2 = a 2 b 2 , and in the second (a 2 d 2 )e 2 = a 2 d 2 - (b 2 d 2 ) = a 2 - b 2 ; . * . the distance from the centre to the focus in any conic represented by the equation x 2 . y 2 is j/a 2 b 2 , whatever value d 2 may have. This is true whether b is real or imaginary. 303. Examples. (1.) Find the nature of the curves represented by the equation (d.) when a 2 = 4, b' 2 = 1, and d 2 = successively 2, 1, 0, with the upper sign, and 1, 2, 3, 4, with the lower sign. (2.) Find the vertices, etc., of the hyperbolas conjugate to those of Art. 287. LINEAR EQUATION OF THE HYPERBOLA. 115 , Proposition 6. 304:. Theorem. The equations p l = ex a and p 2 = also represent the hyperbola x* v 2 in which pi and p z are the focal radii vectores. For, since (Art. 239) p = ed and S = x CO = x -, when the origin is at C, the sign + or - being used, accord- ing as the focus and directrix at the left or right of C is em- ployed to describe the hyperbola, = e / a\ I x - 1 . * . pi = ex a, and p 2 = ex + a. \ e / 305. Schol. Sub- tracting, /? 2 PI= %a, is the equation of the hyperbola in focal co- ordinates. This equa- tion enables us to con- struct the hyperbola by continuous motion as follows : in a piece of thread many times longer than %a make a loop so that its knot shall divide the thread into two segments whose difference is 2a, and fix its extremities at f\ and F* so that I\J? 2 = 2ae. Then, if both seg- ments of the thread slide in a notch near the point of a pencil at P, the .loop L being held in any direction such as to keep both parts of the 116 HYPERBOLA AND ELLIPSE. string taut, and paid out as the pencil advances, then the hyperbola is described. For, F^PL is one segment, and F^PL the other, and by con- struction F 2 PL F l PL = 2a, .' . F 2 P F l P=2a, or p 2 p l = 2a. Proposition 7. 306. Theorem. The equations p l a + ex and p 2 =a also represent the ellipse or a For, as in Art. 304 p = ed and d = - x, . * . p = aex. 307. Schol. Adding, p l + p 2 = 2a is the equation of the ellipse in focal co-ordinates. To construct the ellipse by continuous motion, tie the ends of a thread whose length is 2 ( a + -], and putting it over pins at F\ and F 2 , let the thread run in a notch at the point of a pencil held against it, as at P, the pencil will by its motion describe the ellipse. For, the length of the thread -, and = 2a, or Proposition 8. 308. TJieorem.The equation x+y = i/a z + b 2 represents a line passing through the focus of an hyperbola and the focus of its conjugate. Also the equations and y represent the directrices of an hyperbola and its conjugate. EQUATION OF DIRECTRICES, ETC. 117 For (Art. 284), Also (Art. 300), .-. (Art. 95), A"= CF, = ae, = -/ & = CE l = be 2 = i/o + (+ = 1 represents CL Again (Arts. 281, 106), x = is the directrix 3? ifi irectrix of + = 1 d . . (Art. 284), x = and similarly, y = is the directrix of CL + TJ = 1, 309. Schol. Add these three equations with the signs of the first changed, and they vanish identically ; . * . (Art. 116), the directrices intersect upon the line F l E l at J^, and the parallelogram of directrices is inscribed in the square F^E^F^E^. These relations enable us to construct the vertices, foci and directrices of an hyperbola and its con- jugate ; for aei = b&z = CF l CE^ = A^B^ and CO,: CA L : : CA l : OF, -i.e.,-:a::a:ae l give all the relations necessary. 118 HYPERBOLA AND ELLIPSE. 310. Example. Construct the foci, directrices and vertices of an hyperbola and its conjugate, when a 2 + b 2 = 18 and ab = 6, a and b being the transverse axes of the hyperbola and its conjugate respectively. Proposition 9. 311. Theorem. The equation IE* , _M_ = / a 2 6 2 ~ represents a right line tangent to the ellipse or hyperbola in which Xi and 2/1 w& the co-ordinates of the point of tangency, and the origin is at the centre. A X For, the equation SO is (Art. 90) that of a line through P 2 and P 3 . If these points are upon the curve, then (Art. 295), r 2 , , 2 2 ?/ 2 2 j 2 -v i 3 .^3 / are the equations of condition which subsist. Subtract, TANGENT. 119 Substituting this in the equation of the line through P 2 and P 3 Now let then = y2=yi> and 35 = 3, = ^, . . clearing and transposing, etc., \ , 2 _ a 2 6, a 2 =b& 312. Cor. 1. At the point of contact the equation both of the curve and of its tangent reduces to 313. Cor. 2. If y 0, the intercept of the tangent on x is Cu GL CL OC = XQ = . The subtangent in the ellipse is AB -- x l = , ^ . . by Art. 184, and in the hyperbola AB = Xi = Xi Xi these subtangents are numerically equal to that of the circle in which r = a. This enables us to construct a tangent at (rr 1( yO, as follows. For the ellipse, draw from P 2 a tangent P 2 J/1, and join MI and PI ; MiPi is the tangent at J\. For the hyperbola draw M 2 P 2 from M 2 tangent at P 2 , and join MI and PI ; M t Pi is the tangent at PI. 120 HYPERBOLA AND ELLIPSE. 314. Example. Find the equation of the tangents to the curves x 2 y 2 1 - = 1 at the points whose abscissas are respectively 2 and 6. 315. Exercise. Prove that the equation xx! a(x + #0 ^ y?/i = is that of the tangent line when the origin is at the vertex. Proposition W.'f 316. Theorem. The equation y = mx j/a 2 m 2 b 2 also represents a line tangent to the conic x 2 f- t ~ + ^TTT = 1 ', in which m = tan x - xx i/i/ For, the equation -^ + ^ = 1 (Art. 311), zhZ) 2 ^ . b 2 solved with reference to y t is y 1 "i . . . . (e.) ; a y\ y\ , f 7) 2 x x 2 11 2 . ' . (Art. 104) tan^ = - i L =m. and since ^ + T^TT =1, % a 2 a 2 6 2 Substitute this in (e.), .'. y = mx + j/a 2 m 2 zb b 2 } which is called the magic equation of the tangent line. THE CIRCLES 2 , AND 121 317. Schol. 1. Every equation of this form is tangent to the locus 318. Schol. 2. The equations y mx = -|/a'W b' 2 x r~tf and y = --- h \l -- & 2 , ?tt \ w 2 or represent two tangents perpendicular (Art. 123) to each other. Squar- ing, adding and dividing by (1 + m 2 ), we obtain x 2 + y* a 2 & 2 , which is the equation of the circle which is the locus of the intersection of all tangents perpendicular to each other (cf. Art. 258). 319. Schol. 3. The equation y = or my + x = i/V q= b' 2 , represents a line through the focus (Art. 98) perpendicular to the tangent y = mx + j/a' 2 m 2 6 2 , the co-ordinates of the focus being y\ = and x l = ae = -j/a 2 + b' 2 . Squaring, adding and dividing by (1 -f m 2 ), we find x 2 + y* = a 2 ; hence the locus of the foot of the focal perpendicular upon the tangent is a circle whose radius is a. This principle affords a construction of the ellipse and hyperbola similar to that of the parabola in Art. 257. Draw a circle upon the transverse axis, and through either focus draw chords to the circle ; at the extremities of these chords draw perpendiculars to them, and they will be tangent to the ellipse or hyperbola. By drawing a sufficient number the curve may be defined with considerable accuracy. 122 HYPERBOLA AND ELLIPSE. 320. Exercises. (1.) Prove that the magic equation of the tangent line is y = m(x d= a) \/d i m i + b'* when the origin is at the vertex. (2.) Find the length of the focal perpendiculars p^ and p 2 on the tangent, and show that pip 2 b\ Proposition 11. 321. Theorem. The equation xx i , yyi a : represents a right line which is the chord of contact of two tangents drawn to the conic d 2 ' b 2 from the external point (x 1( y L ). For, let P 2 P 3 be the chord of contact of the tangents P^ and From Art. 90, represents From Art. 311, = -^ .... (/) POLE AND POLAR. 123 represents the tangent at P 2 ; and if it passes through P lt x*x, then - Similarly, 2" + ijT/l = ^ Subtract (/*.) from (p.), . ' . ~ ~ 2 ~ +" xx, , X.X a 1 3 . from (e). -= a;-^3 ,y xx, yy, ' ' from (7i.) ~f + TT? = a 2 d=6 2 ' a 2 .b 2 represents P 2 P y Pi is called the pole, and P*P its polar, with respect to the ellipse or hyperbola. 322. Examples. (1.) The semi-axes of an ellipse are 4 and 2. Find the intercepts of the polar of the point ( 5, 6). Q , - (2.) Apply the data of the last example to an hyperbola by using 2-\/l instead of 2 for the semi-conjugate axis. Ans.x Q = -3~, y = f 323. Exercise. When the pole is on the directrix, the polar is a focal chord. Proposition 12. 324. Theorem. The equation also represents the locus of the intersections of the pairs of tangents to the conies J+JL^ a 2 b* ' 124 HYPERBOLA AND ELLIPSE. drawn from the extremities of all chords which pass through any fixed point (a*, 2/1). For, let Pj be the fixed point through which all the chords pass, and let Q^ 2 be any chord .through P r If the tangents at Q 1 and Q 2 meet in some point P 2 , then 4. is the equation of the chord QtQ 2 by Art. 321, and at P 1 this equation becomes Similarly, if the tangents at the extremity of another chord through P! intersect at P 3 , But (Art. 90) /v/v Eliminate as in Art. 321, . * . -y- + ~^f =^ represents P 2 P 3 , which is the locus of the intersections of all pairs of tangents at the extremities of chords through P r PI is called the pole, and P 2 Pa its polar, with respect to the ellipse or hyperbola. POLE AND POLAR. 125 325. Examples. Find the polars of the following points, with reference to the ellipse x* + 4y L = 16, and the hyperbola x* by* = 16. (1.) (2, 1.) Ans. y = -jx + 4, and y = \x - 4. (2.) (#.-7.) (3.) (2, 0.) (4.) (0, 0.) 28 7 Ans. x = 8. . 28 7 326. Exercise. If the focus be the pole, the directrix will be the polar. Proposition 13. 327. Theorem. The equation xx^ 6 2 ^! represents a normal line to the conic ..2 = 1 a" at the point (x ly y^). For, since (Art. 311) i = x x is one form of the 126 HYPERBOLA AND ELLIPSE. oy - /iy C1 1J equation of the tangent line, .'. (Art. 128), * = ~- X X^ X-i is the equation of a line perpendicular to the tangent at the point (x lt yj that is, the normal. 328. Cor. If y 0, the intercept of the normal is and the subnormal = x 1 x l ( 1 -^= ) = x^ = NM. \ a 2 / a 2 Similarly, if x== 0, the intercept is y = y x ( 1 -- ) = CK. 329. Schol. 1 In the hyperbola, by Arts. 283 and 328, FiN= .Fi(7+ CN= ae + #x lt and (Art. 9) F 2 N= F 2 C + CN= ae + e 2 x lf But (Art. 304), F^ = - a + ex lt and F 2 Pi = a + e^. = e. . ' . J* 7 !-^ : J^A 7 ^ : : -FiPi : -F 2 Pi. In the ellipse, by Arts. 283 and 328, F,N= F,C+ CN= ae JVF, = NC + CF S = ae . . By geometry, NP^ the normal of an ellipse, bisects F&F* the internal angle of the focal radii of PI. ,. Also,DP 1 N=NP 1 . Subtract, . . . . the tangent bisects FPiQ the external angle of the focal radii. Similarly, the normal of the hyperbola bisects the external and the tan- gent the internal angle of the focal radii. NORMAL. 127 Conversely, a tangent being drawn to an ellipse or hyperbola, to find the point of contact. Through F* draw JF^Q perpendicular to the tan- gent, and make Q,S=SFv\ through Q draw Qf\, and the point P lf in which it cuts the curve and tangent, will be the point of contact. 330. Schol. 2. f Solve the equation of the normal for y, Let y = b- t 77?- y(a 2 b 2 m 2 ) is the magic equation of the normal. 331. Exercises. (1.) Show that the equation 2/Hl! = _^L- # a?! 6V, also represents a line through the pole (x v , T/J) perpendicular to the polar ^L + -^ = 7, with respect to the curve 4 + ~T, = 1 ( Art - 128 )- (2.) Show that in the ellipse, P 1 N= - i/a* e*x?, and that . . (Art. 306) ^ 2 = P,N . P^. 128 HYPERBOLA AND ELLIPSE. (3.) Show that in the hyperbola also ^/> 2 = P^N Proposition 14. 332. Theorem. The equation x cos y sin + -T^= represents that diameter of the curves a =h which bisects the system of parallel chords each of which makes the angle with the axis of x. For, from Art. 100, if : rr = T = I, then I is the sin o cos u distance from any point (x lt yj upon the line y b l =x tan 0, to some point (x, y) at the intersection of this line with x 2 -* + -*- . . x = x l + 1 cos 6, and y = y^ + 1 sin 6. Substitute these values of x and y, in (be.). (x, + I cos O) sn Expand, cos 2 sin 2 ff \ x, cos g DIAMETERS. 129 There are evidently two values of I. Let these values be equal numerically, but of opposite signs ; then the point (x lt y^ must bisect the chord, and, by the " General Theory of Equations," the coefficient of the second term i. e., the coefficient of the first power of I must vanish ; for the quadratic is the product of the sum and difference of the same quantities. x l cos y.^ sin 6 '' ~~oT~ ~~ ' in which (x v y^ is the middle point only of any chord. Now, making (x lt yj general, x cos 6 y sin 6 _ is the locus of the middle points of all these parallel chords (Art. 242), and is evidently a right line through the origin that is, through the centre. ' 333. SchoL When x l = 0, and y l = 0, we have the distance from 1 /cos 2 sin 2 0yi the origin to the curve - I - | -- 1 , . . /is always real m 6 \ Cb ~"~ U I xi IT i ' i n i i cos 2 sin 2 the ellipse, and is real in the hyperbola when - -- ?. e., when a 2 1 tan < HZ -, but imaginary when tan > - that is, in the latter a a case the line in the given direction does not cut the curve 4+-^ = ^ butdoescut * . a? b* a 2 b* 334. Examples. (1.) The semi-axes of an ellipse are 4 and 3 ; find the equation of the bisectrix of a system of chords making an angle tan ~'i/3 with the axis of x. Am - = - (2.) Find the same for the hyperbola whose semi-axes are 4 and 130 HYPERBOLA AND ELLIPSE. Proposition 15. 335. Theorem. The equation b 2 m 1 m 2 = -- expresses 1st. The condition that the diameter y = m. 2 x shall bisect the system of chords y bi = m^, (in luhich 6 : is a variable constant). 2d. The condition that y = m& and x = m z x shall be con- jugate diameters. 3d. The fact that y yi=m,(x x l ), the tangent at the extremity (x^ y^) of the diameter y = m^, is parallel to its conjugate diameter y = m 2 x, and to the system of chords y = 4th. The fact that supplemental chords y yi=m l (x x^ and y + 7/1 = w* ( x + are Always parallel to conjugate diameters y^i^x and y = m 2 x, 1st. The equation of the diameter (Art. 332), x cos 6 y sin 6 -- L* - a 2 b 2 ' b 2 solved for y, is y ~ T" x cot #> or y = mp. The equation CJL of the chords it bisects is y = x tan + b l} or y = m^x -f b lt b 2 2d. "When y = x tan 6 -\- b l} or y = m-p + b v is the system b 2 of chords, then (by 1st), y = x. -7- - 7 , or y = m 2 is their diameter i. e., the coefficient of x for the diameter is obtained from the coefficient of x for the chord by multiplying its reciprocal by y . CONJUGATE DIAMETERS, ETC. 131 b 2 Similarly, when y = x' -JT ^ + b. 2 , or y m^c + 6 2 , ci" tan is the system of chords ; multiply the reciprocal of the coefficient of x by - = Y~, . * . y = x tan 0, or y = m^ is the diameter, which is parallel (Art. 128) to the chords y = m^x + b lt . * . (Art. 242) y = m l x and y = m^c are conjugate, 6 2 3d. Now, y y\ = ~~i l (x x^, or y y x = m 2 (x arj rep- resents (Art. 317) the tangent at (x lt yj, and y = x, or y = m l x, 1 is the diameter through (# 1; yj, 4th. When P is any point, y y l = m l (x x^ represents PP r Also (since y 2 = y^, y + y x = m 2 (a? + a^) represents PP 2 . Multiply, . ' . y 2 y x 2 = m^j (or 2 a^j 2 ) . . . (m.) x 2 y 2 Again, if P l and P 2 are on the curve, then -y + ~~f* l t x 2 y 2 + = / a 2+ 6 2 2 ' and if P or (x t y) is also on the curve, Subtract, .'. - ~--f- ^-=^ or y 2 -y 1 2 = -= 1 ^- (a; 2 -^ 2 ), a 2 b 2 a 2 which (Art. 112) is the equation of the two chords PP l and PP 2 respectively. Divide the equation by equation (m.) 9 b 2 132 HYPERBOLA AND ELLIPSE. 336. Schol. The conjugate diameters in the ellipse both cut the curve (Art. 333), and since m^m^ = -- -, if m is the tangent of an acute angle it is + , and m 2 is , and is then the tangent of an obtuse angle, and vice versa. In the circle a = b, . . m^n* = 1 (cf. Art. 123). The angle between the axis of x and the conjugate diameters in the hyperbola are both acute, or both obtuse, since nfi^m^ = -\ -, and if Wi > , * - wi 2 < -. Hence (Art. 333), only one of the con- a a jugates intersects the hyperbola. x* y z 337. Examples. (I.) Given an ellipse -- \-- = l t and a diame- J.O t/ ter making an angle of 30 with the axis of x ; to find its conjugate. Ans. 16y = (2.) Find the equation of a tangent to the same ellipse parallel to the line ---1 = 1. Ans. y = 2x 8.644. o.o 7 X* 7/ 2 (3.) Perform the same operations with the hyperbola * = 1. It) a 338. Exercise. Construct conjugate diameters; also a tangent parallel to a given line. Proposition 16. 339. Theorem. The equation represents an ellipse or hyperbola referred to conjugate diameters as co-ordinate axes; in which a? and :b* are the squares of the semi- conjugate diameters. For, change the direction of the axes in the curve ^ + 7 2= ^' By Art. 71, x x f cos 6 + y r cos 1; and y = x' sin 6 + ?/' sin O r CONJUGATE DIAMETERS AS AXES. 133 (x r cos 6 + y' cos O^ 2 (x r sin + y r sin 0^ *> .TO or Y' Y But since these are conjugate diameters, by Art. 335, b 2 cos 6 cos 6 l sin sin 6 l tan(9 tan^ + 7- = ^, .'. by trig.- -^ + ^ (2 y cos 2 sin 2 7 cos 2 0, sin 2 ^ 1 = ;, and ' ' But(Art.333),^^- + ^- = - 2 , x' 2 , y' 2 340. Exercise. Prove that -^-+ ^ =-? is the equation of the i 2 b? tangent referred to conjugate diameters. Proposition 17. 341. Theorem The equations Til Til ^l , a b together represent an ellipse; in which

. 136 HYPERBOLA AND ELLIPSE. 346. Schol. 2. Since in the parallelogram we have This principle is employed in the construction of the trammel or elliptic compasses. For, if MI and N run upon the axes as guides, any point P! of the line MiN will describe an ellipse. The point halfway between M and N describes a circle, a particular case of the ellipse. 347. Schol. 3. The point P! corresponding to the eccentric angle

, we have by subtraction Hence, draw two concentric circles whose radii are respectively a and b, and from the points Q v and N where any radius cuts the two circles, draw parallels to the axes of y and x respectively as repre- sented ; their intersection will be a point of an ellipse whose semi-axes are a and b. /v fit 348. Exercise. Show that - cos

i=l is the equation a b of a line tangent to the ellipse at a point whose eccentric angle is ? lt Proposition 21.^ 349. T7ieorem.The equations x y - = sec a> and 7 = tan

, must represent the same. ECCENTRIC ANGLE OF THE HYPERBOLA. 137 350. Schol. 1. The eccentric angle of P is XCP Z . For, if # 2 7/ 2 <7P 2 = , then , by trig. Also, if CP 3 = , -=tan then . . also P 3 M 3 = 351. Schol. 2. The eccentric angle affords a method for constructing the hyperbola by points. Draw two concentric circles with the radii a and 5, and from the points where any radius cuts the two circles draw the tangents P 2 M, P 3 M 3 . Then M^N=M,P z =y and CM=x. Through M and N draw parallels to the axes y and x respectively ; their intersection at P is a point of the hyperbola. 352. Exercise. Show that the line x - sec a , tan 4" Q T" /fl A/t/i //t/9 9 9 Ltlll U/| Lclll 1^9* a^ a j i . . tan ^ tan 2 = l. . . (Art. 123), ^ x y? 2 = ^- /^Y /^\2 T y 2d. In the hyperbola (-) (7) =-?, L =sec p, and 7 = tan CP. ^ r \a/ \b/ a b Similarly in the conjugate hyperbola TTJ (-J =1, r = cosec 2 . 2 x 2 a cot tf 2 a For, if y = ra 1 a; cut the primary hyperbola, then y=m 2 ; sin =/j .-. by trig. CONJUGATE DIAMETERS. 139 355. Schol. This enables us (see fig. of Art. 347) to construct con- jugate diameters in an ellipse by drawing two radii of the circle r = a at right angles. The ordinates of their extremities Qi and Q. 2 will give the extremities PI and P. t of the conjugate diameters. Conjugate diameters in the hyperbola may be constructed as follows: Draw two circles with a and b for radii, and let the angle nd P. 2 , having the same and B^, will be the XOY= XCQ l + XCQ* = -. The points P^ and P. 2 , having the same subtangents M^M^ and JV^ as the arcs extremities of the conjugate diameters. Proposition 23. 356. Theorem. The equation a l b l sin p = ab expresses the fact that the tangents through the extremities of any pair of conjugate diameters enclose the same area ; in which ft = o l 0^ is the angle between the semi-conju- gate axes di and bi. Let j- + ~h = 1 be the tangent line through (x l} y^, the extremity of the diameter y = i a 2 ;, and parallel to the diameter 7 = 0. 140 HYPERBOLA AND ELLIPSE. By Art. 142, the length of the perpendicular from the origin on the tangent is p= But (Arts. 341, 342), Now, in the figs, of Arts. 347 and 355 the area of the parallelo- gram CP^DPz =pb lt but phi = ab. Again, by trig., CP 1 DP 2 = a 1 ^ 1 sin /9 . . ab = a^ sin /?. Now 4i sin ft = rectangle DD' . 357. Schol.-Ky Art. 356, sin' - ~ = -,^ A 2 , 2 . + &i 2 ) 2 - (i 2 - V) 2 ' . . (Art. 342) sin 2 /? = I , _ 2 for the ellipse. . . (n.) the In the ellipse, when a : = Jj, sin ^ ~ ~ is a minimum value of (.). These are the equi-conjugate diameters ; and since the ellipse is symmetrical, Wj = m 2 in the equation m 1 m 2 --= -- ^, CL and .*. w= for the equi-conjugate diameters. .*. they lie a in the diagonals of the rectangle of the axes. In the hyperbola, sin = is a minimum value, for a^ increases with bi, and the denominator of (o.) can be made infinite. These conjugates b' z are infinite, and coincide. Since m^m^ = , a 2 when m l = m. 2 , then m=-. These are self-conjugate diameters, and will be shown to be asymptotes i. e., tangents at an infinite dis- tance from the origin. Also (Art. 284), e = sec A l CD l (fig. of Art. 358). ASYMPTOTES OF THE HYPERBOLA. 141 Proposition 24. 358. TJieorem.TJie equation represents the asymptotes of an hyperbola and its conjugate hyperbola. For, the equation of the tangent line . g x^ + yy^ =L which is the inter- cept on x. If x = 0, y n -, which is 2/i the intercept on y. If the co-ordinates of the point of tangency are 0^ = 00 and y^ = oo ; then X Q = and y Q = 0. . . the tangent at a point infinitely distant passes through the origin. Also (Art. 342), of = x? + y?, .' . a l = oo ) and then (Art. 357) b b m = :-, .'. y = mx, or y = -/is at once a diameter and tangent at infinity. The same may be proved for the conjugate hyperbola. The equation y = -x may also be written (x y\ fx y\ x z v 2 -I ~ + I )=, o r -ij^ - a b/ \a b] a 2 b 2 359. Schol. The asymptotes are the diagonals of the rectangle formed by the tangents at the vertices i. e., the rectangle of the axes. 360. Exercise. Prove that y = x is the equation of the a l asymptotes referred to conjugate diameters as axes. 142 HYPERBOLA AND ELLIPSE. Proposition 25. 361. Theorem. The equations xy = -j- represent an hyperbola and its conjugate referred to the asymptotes as axes. x' . . cos = X and . sin x --sin ; . . (-Art. 71) x (x' + y r ) cos d and y = (y'-x f ] sin 6. But (Art. 357) tan = -, . ' . by trig., cos = , 2 . ~T ) and = b a(x'+y>} hence ^ =2 8 d2/ - a: 2 y 2 Substituting these values in jr = we have (x' + y'Y (y'-xj 1, whence a? Similarly, substitute in 2 + TJ =.? for the conjugate hyperbola, 4 ' ASYMPTOTES AS AXES. 143 X I/ 362. Schol. 1. The equation h = 2 represents the tangent at (a? lf yj) referred to the asymptotes. For, the line through (a:,, yj and (a? t> y 2 ) is (Art. 90), y \_yi- y\ But y l = a:* y 2 = y yi a; =.2L. Place *, = :*, then - + - = *. 363. Schol. 2. In this, if y = 0, x Q = 2x l = intercept on x ; and if x = 0, y Q = yi = intercept on y. . . a: y = 4x\y\ = a 2 + 6 2 ; . ' . the product of the intercepts is constant. 364. Schol. 3. By trig., area CP l = x^i sin = - sin 20, since - 4 = 2 sin cos e =~ (Art. 361). . * . area GA\ = area CPi = . 365. Schol. 4. Since x^ = and yj = , the point of contact (by sim. / C M I K tri's) bisects JOT, the portion of the tangent between the asymptotes. Since OP l bisects MN, it bisects any parallel as HK at Q. . . HQ = QK. But since CP } is a diameter, PQ = QA t . . ' . HP = A^. Hence to construct an hyperbola by points, draw a series of lines through A, and in each line make HP=AiK, then the locus of P is the hyperbola. 366. Examples. Construct the loci represented by the following equations. 144 HYPERBOLA AND ELLIPSE. (1.) + +- = 1, in which = 60. JL ts (2.) 4x* y* 4, in which 30. Find the equation of a tangent to each of the above loci at a point whose abscissa is 2. Ans. (1.) (20 4x (3.) (40 (5.) 17x = 68. 367. Exercise. Prove that the asymptotes are the diagonals of the parallelogram formed by drawing tangents through the vertices of any pair of conjugate diameters. Proposition 26. 368. TJieorem.The equation I ' 1 + e cos 6' represents an ellipse or hyperbola ; in which p and 6 are the polar co-ordinates, being measured from the nearest vertex, and- I is the semi-latus rectum, the pole being a cut one of the foci. For, in the ellipse (Art. 307), Pi + p 2 = % a > and by t- ri g- PI = Eliminate = 1//? 2 2 Squaring and reducing, cos u. p2 **a , cos 6. b 2 e cos 6 a 1 + e cos (Art. 288). POLAR EQUATION. 145 Let = -, then, f> 3 = =1, .'. p = a 1 + ecosd' The same may be proved for the hyperbola. 369. Schol. The equation p = -j represents an ellipse, and the equation p = = represents an hyperbola, in which is 1 e cos measured as usual from + x, and the sign -f or is used, according as the right or left hand focus is the pole. 370. Examples. Construct the curves represented by the follow- ing equations. w (2.) r.q ^ 3 + i/5 cos 1 2 + tan - cos 0. S 4 3 + cos + 4 cos sin Construct the polar of the point (4, 6) with reference to the follow- ing loci. (4.) x*-y* = (5.) x* + y 2 = 4, in which a> = V = 60. Ans. (6.) Interpret the equation x* y 2 = x + y. 371. Exercises. (1.) Prove as in the parabola that the focal polar equation of the tangent line is /> = - ; - -, in which is e cos + cos (0 9$ measured from the nearest vertex. (2.) Prove by transformation that the central polar equation of the ellipse and hyperbola is p = - ^ - - . 10 CHAPTER VIII. GENERAL EQUATION OF THE SECOND DEGREE. Proposition 1. 372. T7ieorem.The most general equation of the second degree, viz. : Ay? + 2Hxy + By 2 + %Gx + 2Fy + C= 0* . . . (a.) always represents one of the conic sections when A, B, C, F, G and H are any real constants. The truth of this proposition will appear as the result of the succeeding propositions, in the following manner. We shall move the origin either to the centre or to the vertex, and then change the direction of the axes of x and y so that the axis of x shall coincide with the principal axis of figure of the curve ; it will then appear that (.) is reduced to a form identical with one of those before found to belong to the conic sections. 373. Schol. 1. The same result would follow whether (a.) be in oblique or rectangular co-ordinates. For, if it be in oblique co-ordi- nates, and be transformed to rectangulars, the co-efficients A, B, C, etc., will be changed, but the general form will remain the same (Art. 229). We shall, therefore, without the loss of generality, consider the case of rectangular axes. 374. Schol. 2. In this discussion A is supposed to be a positive quantity. * We here adopt the coefficients in ordinary use at present. x It may be useful to notice their symmetry as expressed in y the square ; e. g. t B is the coeff. of y 2 , 2F of y, 2G of x, etc. 1 146 CO-ORDINATES OF THE CENTRE. 147 ^Proposition 2. 375. Theorem. TJie equations BG-HF AF-HG express the co-ordinates of the centre (x , y ) of the locus represented by equation (). For, let us move the origin to (x , y )> whose co-ordinates are yet to be determined. From Art. 23, x x i XQ } and y i . . (a.) becomes - G)x f - Now, if (X Q , T/O) is the centre, (6.) must be of such -form as to remain unchanged, whether we substitute for x' and y', + x' and + y', the co-ordinates of one extremity of a diameter, or x r t and y', the co-ordinates of the other extremity of the same diameter, since then, the origin evidently bisects the distance between (x' ', y'} and (x/ y'). That such may be the case, there must be no terms of the first power in (6.) i. e., from which by elimination we find _BG-HF ^AF-HG- X ~ H 2 -AB'- (Ct}> and 2A>- H 2 -AB '" 376. Schol. l.li H 2 A = 0, the centre of the curve is at infinity i. e., it has no centre ; but if JET 2 AB>0, it has a centre. 377. Schol. 2. It may he noticed that the transformation result- ing in (6.) does not change A, B or H, and that the new constant term Axfa-QEx&t+Byfa Gxo+2Fy*+Q-= C r . . . (e.) is of the same form in X Q , y as (a.) is in x, y. 148 GENERAL EQUATION OF SECOND DEGREE. 378. ScJiol. 3. Hence the result of the transformation to the centre may be written 379. Examples. Find the co-ordinates of the centres of the curves given in Art. 391, and the values of G'. Proposition 3. 380. TJieorem.The equation A-B expresses the value of the angle through which the co- ordinate axes x and y in equation (a.) must be turned to cause them, to be parallel to the axes of the curve. To turn the axis through any angle, 6, we place (Art. 80) x=x" cos 6 y" sin 6, and y = x" sin 6 +y' f cos 0. Substitute in (a.) . ' . (A cos 2 6 + J7sin 6 cos 6 + B sin 2 0) x" 2 + 2[(B - A) sin 6 cos 6 + ^T(cos 2 6 - sin 2 6}] x" y" + (A sin 2 &5"sin 6 cos 6 + B cos 2 6) y" 2 >-sin This may be written A V' 2 + 2HWy" + B'y m + 8G'x" + 2F f y"+C= . . Now, if H' = 0, or H(cos?0-sm 2 d)- (A -B} sintfcostf =0, H sin 6 cos# then, -B cos 2 0-sin 2 0' AXES OF FIGURE. 149 By trig., sin 20=2 sin 6 cos 0, and cos 26 = cos 2 sin 2 0. H j sin0 i Substitute, . . T7~ = * ' " "^ = * tan ^ < (*) J\. x3 COS /v " Hence (ft.) becomes ^V' 2 + y' 2 + V'+^y' + <7=0. . . . (I.). Now complete the square with respect to both x" and y". or '\ 2 \ _i_ 7?/ ? /r j __ _i -- C= (y '} H \ y ^ 5V ' ' ; G^'V / .F\ a ^-) r + ^ which (Art. 296) represents an ellipse or hyperbola referred to axes parallel to the axes of the curve, when neither A' nor B r is equal to zero. 381. Schol. 1. By a similar transformation of (/.) its axes of x and y will become coincident with the axes of the curve ; call the result ( 0. For, when tan 0, .'. (r.} H 2 -AB> 0. If, however, -A'B'=H 2 -AB = 0, then A' = 0, or B'=0. Suppose A' 0; by Art. 376, the centre is in this case at infinity, and equation (I.) becomes B'y' f2 + QF'y" + 2G- f x" + C= 0, I jF'\ 2 BG-'I C F' 2 which (Art. 243) represents the parabola y 2 = - -^7 x, whose origin has been moved by using (Art. 23) equations, 152 GENERAL EQUATION OF SECOND DEGREE. B'C-F' Z F f when x = ~ 7r ~ 2/o~7- 388. Schol. l.li A'B = H*-AB<0. C' C' When -- -r,>0, and - > 0, (n.) represents a, real ellipse (Art. 288). A JD When C' = 0, the ellipse reduces to two imaginary right lines. C' C' When -- < 0, and < 0, the ellipse is imaginary (Art. 301). A. B If A' = B', the ellipse becomes a circle. 389. Schol. . If A'B = H' i -AB = 0, and A' = only, (s.} represents a real parabola. If also G r = 0, then, when F n B'C^> 0, the parabola becomes two real parallel right lines ; when F'* B C= 0, the parabola, becomes two real coincident right lines ; when F' z B'C 0. C r C' When -- - > 0, and < 0, the hyperbola is primary (Art. 281). ^1 JD When C' 0, the hyperbola becomes two right lines. C' C' When <, and -- > 0, the hyperbola is conjugate (Art. 299). A B If A' B', the hyperbola is rectangular. 391. Examples. Show what curves are represented by the follow- ing equations. (1.) r 2 + 4xy + f + 3x + 2y + l = 0. (2.) 2x 2 + 2xy + 2y' 2 = x y l. . (3.) y 2 + 8xy-3x = 0. (4.) CHAPTER IX. CURVES OF THE THIRD AND FOURTH DEGREES. THIRD DEGREE. Proposition 1. 392. TJieorem.The general equation of the third degree + A, has been shown by Newton* to be reducible in all cases to one of the following four forms : xy = x + x 2 +Cx + D .... (c.) y 2 = Ax* + x 2 +Cx + D .... (d.) y =Ax 3 + Bx 2 + Cx + D . . . . (e.) Included under (&.) are a large number of curves of various forms, all having at least two infinite branches. This may indeed be proved to be true of all curves of odd degree by Art. 215 ; for a curve of the n th degree is cut in n points by one of the first degree that is, by a straight line ; and if cut in an odd number of points by a straight line, it cannot be composed entirely of finite loops. *See Newton's " Enumeration of Lines of the Third Order." 153 154 CURVES OF THIRD AND FOURTH DEGREES. Equation (e.) embraces but one form, called the trident, and equation (e.) represents one form only, called the cubic parabola. Those included under (d.) are of five species, and are called parabolas. It has been shown that the shadows of these five parabolas that is, their conical projections upon planes situated in various positions will give rise to all the other curves represented by (a.). Proposition 2. 393. TJieorem.The semi-cubic Parabola P is the locus of P, the intersection of the ordinate QA of the parabola if = 4px with that perpendicular OM to the tangent QM, which passes thi*ough the vertex 0. For, if x l and y l are the co- ordinates of Q, and x' and y' of P, then (Art. 128) is the perpendicular through on the tangent (Art. 249) and x=x l = x f . . . ( and XCP = OPC + COP = 3COQ. .'. in the triangle COP p:2a:iBm30: sin 20, sin 3d .-. ^ = ^a gin 0' which is a polar equation of the curve. If A NQ = a, then \T .- . p=4 a cos a sec #, which is another form of this polar equation. By transforming either of these polar equations by Art. 86 we obtain the above rectangular equation. 401. Schol. Let a line through (7 cut the curve in P, P' and P", then P'OP=P"OP' = 60*. For, taking p in any position, as OP, if XOP=0 = O l , then XCP =30-, and if a new position of ^ be taken, as OP', such that = 0, 60*, then XCP ' = 5 (0 l 60) =30 l 180 that is, CP is in the same right line with OP'. Again, if another position of p be taken, such that = 1 : 120, then XCP" = S(6 l 120) = 30 l 360 i. e., we have still the same line. 402. Exercise. Show that the polar equation of this curve with the pole at is P a sec J0, and that the equation of the curve when the axes are the tangents OS and 02 7 (in which XOS=60*) is + y'* = 0. Proposition 6. 403. Theorem. TJie Folium is the locus of P so situated with reference to two points A and that (if A0 = 3a) then ON=20M = 2x, and OQ = AM= 3a + x, and also MOP --= POQ. 158 CURVES OF THIRD AND FOURTH DEGREES. For, then ^ = cos 20 ; 3a-\-x . by trig., 404. Schol. This is a projection of the trisectrix in which y = y'~\/3. 405. Exercise. The equation referred to the tangents 0$and OT as axes (in which XOS=45) is of the form = o. Proposition 7. 406. TJieorem.The Logocyclic curve 2a-x has the product of its two radii vectores constant. For, transforming to polar co-ordinates by the equations x = p cos d, and y = p sin 0, transposing, factoring and reducing by the relation sin 2 6 -f cos 2 = 1, On we have cos u whence p a(sec tan 0) ; . ' . p,p 2 = a 2 (sec 2 6 - tan 2 0) = a 2 ; . . also OA = a. This curve has a parabola as the locus of the intersections of the normals drawn at P! and P 2 . FOURTH DEGREE. 159 407. Exercise. The locus of P when OP= OB + OJ/is (if OA = a) p = a (sec + cos 0), which is a curve of the third degree. This is one of the family p = a (sec + n cos 0), to which belong the cissoid and trisectrix. FOURTH DEGREE 408. There are some thousands of curves of the fourth degree, a few of the more noted of which are discussed in the following propositions. Proposition 8. 409. Theorem. The Lemniscata or p'p" = = a ' cos 20. 411. Exercise. Show that the polar equation of the Ovals of Cassini, which are denned by the equation 160 CURVES OF THIRD AND FOURTH DEGREES. in which p' and p" are focal radii and b is any constant, is when the pole is halfway between the foci, whose distance apart is 2c. Proposition 9. U2. Theorem. The Limacon is the locus of P at the intersection of two lines OP and AP moving so that XAP=\XOP. For, in the triangle AOP, if OA = a, then p : a : : sin |0 : sin ^( ^ =^ a (^ -j- ^ CO s 0), which equation, transformed by Art. 86, gives the above equation. 413. Schol. 1. The equation p a -f ^acos shows, since ON= 2a cos tf (Art. 197), that NP=a. 414. Schol. 2. It may be shown in a manner similar to that used in Art. 401 that POP, = P.OP, = Q0\ 415. Exercise. (1.) Show that the polar equation of the limacon with the pole at A is p 2a cos J0. (2.) Construct the figure for the more general equation of the lima- fon, of which the foregoing is a particular case, viz. : CONCHOID. PEDAL CURVES. 161 or p =p cos a ; when p^> a and when p t, but the axis of y tangent at the origin when r t and hence T 1 > t 1, then, by algebra, 4n = 0, . ' . equation (a.) becomes y =0, y\ 163 164 HIGHER ALGEBRAIC CURVES. . ' . (Art. 106) this tangent at the origin is the axis of x. Whenever r < and hence r 1 < 1, then by algebra . . (Art. 106) this tangent at the origin is the axis of y. r>t r t or r < t, the former having a cusp at the origin. 425. Schol. 2. When r is odd and t is odd, the curve is symmetric in oppo- site quadrants (Art. 238, 2d), and there are two forms according as r > t or r < t, both having a point of inflexion at the origin. 426. Schol. 3. When r is even and t is even, the equation may represent at least two separate curves. For, the equation, when both members are positive, takes the form y 2H = 6V" 1 . Transpose and factor, . * . (y n -f- &c m ) (y n bx m ) = 0. .'. by algebra, y = bx m and y n = bx m . But when one member is positive and the other negative, the curve is imaginary. When r = t, the equation represents one or more right lines. Proposition 2. ' 427. Theorem. Hyperbolic Curves represented ly equations of the form y*x r = a, in which r and t are positive in- tegers, all have both the axes of x and y tangent to them at infinity i. e., both are asymptotes. PARABOLIC AND HYPERBOLIC CURVES. 165 For, at any points (x 2) y 2 ) and (x 3 , 3/3) upon the curve we have 2/2* = ax 2 ~ r and y$ = ax 3 ~ r . Subtract, . * . y 2 * yj = a(x^ r x 3 ~ r ) = t j2 j& V 2 2 3 2 3 ' *^2 "^3 *^2 *^3 \2/2 2/2 2/3 ^2 2/3 Substitute in the equation of Art. 90, and then let T y-y\ ar xx, r+l But 2/1 - ^-jf (6.). re ^ ^ Equation (5.) represents a tangent to y t x r = a at ( In (6.) let a?! = oo, then y^ and - 77 = 0, , * . (Art. 106) this tangent at (oo, 0) is the axis of x. Again in (6.) let y l oo, then x 1 = and - 1 = oo, . * . this tangent at (0, oo) is the axis of y. 428. Schol. 1. When r is odd and t is even, the curve is symmetric about the axis of x (Art. 237, 1st). 429. Schol. 2. When r is odd and t is odd, the curve is symmetric in opposite quadrants (Art. 238, 2d). 166 HIGHER ALGEBRAIC CURVES. 430. Schol. 3. When r is even and t is even, the equation may represent at least two separate curves (Arts. 236, 426). 431. Schol. 4. It appears from the nature of rectangular co-ordi- nates that when in the equation of any parabolic or hyperbolic curve discussed in this or the preceding proposition, y is written for x and x for y, the curve is thereby revolved 180 about the line whose equa- tion is x=y i.e., about the bisector of the first angle. But if y and x be written for x and y respectively, the curve is thereby revolved 180 about the line x y i.e., the bisector of the second angle. If y be written for y, the curve is thereby revolved 180 about the axis of x ; but if x be written for x, the curve is revolved 180 about the axis of y. Any combination of these replacements may be effected by perform- ing them successively (cf. Arts. 237, 238). ^Proposition 3. 432. Theorem. 1st. If a given curve has one or more branches through the origin (Art. 213), either a parabolic curve or right line may be found, one for each branch, which also passes through the origin, and which in shape and direction approximates to the branch near the origin. %d. If a given curve has infinite branches, either a para- bolic curve, hyperbolic curve or right line may be found, one for each branch, which approximates to the position and direction of the branch at infinity. This proposition is one of the general results of succeeding propositions in this chapter. 433. Schol. By moving the origin to different points of the given curve, the shape and direction of the curve may then be found by means of its less intricate approximate curves. Hence the approximate curves depend on the position of the origin. APPROXIMATE CURVES. 167 EXPONENTIAL POLYGON. 434. The exponential polygon * is a device which enables us to find readily which of the terms in the equation of a curve of high degree may be neglected for the purpose of obtaining each of the simpler equations which represent curves approximating to the different branches of the given curve. 435. Exponential Axes. Draw any two lines OR and OT at right angles as exponential axes ; and using the exponents r and t of any term ax'y* as the exponential co-ordinates, locate with respect to OR and OT the representative point (r, t) ; and in the same manner locate a representative point for each term of the given equation by using its exponents as co-ordinates. E. G. In the equation (y 2 - ax? (x - of - a?xy (x* + y* - a 2 ) = 0, or % ay 2a?xy* + aV aVy aV?/ 3 + ctxy = 0, R the term a 2 ?/ 4 has the representative point (0, 4) marked by a small circle on OT. The term Batfy* has the representative point (3, 2), * The exponential polygon here used is in principle the same as the " ana- lytical triangle" and "parallelogram" employed by Newton and others. 168 HIGHER ALGEBRAIC CURVES. and the other terms have their representative points as shown by the small circles marked upon the figure. 436. Remark. It will be seen that we have thus effected an arrangement of the terms of an equation in rank and file according to the powers of x and y in those terms, and that the representative points show only this, viz. : that terms containing certain powers of x and y occur in the equation ; they in no sense represent points of the curve. In locating the representative points all coefficients are dis- regarded, since they in no way affect the position of those points. 437. Definition. Draw the smallest convex polygon, having representative points at its corners, which can contain within it or upon its sides all the representative points : this is the exponential polygon. 438. Exponential Equations. The sides of this polygon are straight lines, and may be conveniently represented by expo- r t nential equations such as - + 7 = 1, which will facilitate our dis- cussion of the polygon. The equation of the curve is assumed to contain no negative exponents, hence no representative points are on the negative side of either of the axes of r or t. There may be fractional exponents in the given equation, though in the cases we treat we shall assume the exponents to be integers. It is also assumed that neither x nor y enters every term of the given equation ; for if every term contain x, for instance, the equation is exactly divisible by x t and that factor may be removed. Since those terms which do not contain y have their repre- sentative points on OR, and those which do not contain x have theirs on OT, it is evident that the polygon has either a corner or a side in each of the axes of r and t. 439. Sides. If each side of the exponential polygon be pro- duced indefinitely, five kinds of sides may be distinguished, as follows : 1st. Sides whose intercepts on the axes of r and t are both positive, and which also lie between the origin and the rest of the polygon. APPROXIMATE EQUATIONS. 169 2d. Sides which have both intercepts positive, and which have the rest of the polygon between themselves and the origin. 3d. Sides whose intercepts are one positive and one negative ; and also sides through the origin, all parallels to which have inter- cepts, one positive and one negative. 4th. Sides that coincide with either of the axes. 5th. Sides parallel to either of the axes, which have the rest of the polygon between themselves and the origin. 440. N.B. If now from the given equation all terms be omitted except those whose representative points lie upon a single side of the exponential polygon, let us call the result " the approximate equation fop that side." And if from any equation all terms be omitted except those whose representative points lie on any assumed right line, let us call the result " the approximate equation for that line." 441. The following proposition will be shown to be another of the general results of the succeeding propositions of this chapter. Proposition 4. Theorem. Each approximate equation for a side of the exponential polygon represents an approximate curve. When the approximate equation is for a side of the first kind, the approximation is near (0, 0). For a side of the second kind, the approximation is near (oo, oc). For a side of the third kind, the approximation is near (0, oo) or (oo, 0). For a side of the fourth kind, the approximation is an inter- section of the curve with the axis of x or y. For a side of the fifth kind, the approximation is at infinity, and to a right line parallel to the axis of x or y. Or, as it may be stated in other words, each approximate equation for a side of the exponential polygon represents an 170 HIGHER ALGEBRAIC CURVES. approximate curve, when x is made infinite in all approximate equations for right-hand sides, and infinitesimal in those for all left-hand sides ; while at the same time y is made infinite in all approximate equations for upper sides, and infinitesimal in those for all lower sides. 442. E. G. In the example, Art. 435, for sides of the first kind the equations are (see figure) a\f + cfxy = 0, or y* a?x, and a*x 2 + cfxy = 0, or y = x, which represent, as will be shown, curves approximating to the branches of the given curve near the origin. There is one side of the second kind from which #y 2ax\f + aV = 0, whence y 2 = ax, which represents a curve which approximates to one of the infinite branches of the given curve. There is one side of the fourth kind from which aV = 0, whence x = a, which is the intersection of the curve with the axis of x. There is one side of the fifth kind from which y 2axy* + #y = 0, whence x = a, which is the equation of a straight line which approxi- mates to one branch of the curve near (a, oo). Proposition 5. 443. TJieorem.TIxe approximate equation for any assumed right line is of the form (r and t bei7^g positive integers), whenever the intercepts on that line upon the exponential axes are both positive; but is of the form y V a ivhenever they are one positive and the other negative. POSITIVE AND NEGATIVE INTERCEPTS. 171 T t For, let -- \-~ -=1 be the equation of the assumed line r o c o referred to the axes of r and t, in which r and t are the inter- cepts. If r l and j are the exponents of one term of the approxi- mate equation for this line, r 2 and t 2 those for another, r 3 and 3 those for another, etc., then we have the equations, T l , ^1 y r 2 , ^2 y r 3 , ^3 y ~ + T~ 1, 7 + 7~ A r "" T~ = r o ^o r o J o r o c o Subtracting, transposing, etc., which is the relation between the exponents of the terms whose representative points lie on this line i. e., the relation of the exponents in the general form of the approximate equation for this line. This general form may be written aXY l + a^Y 2 + a 3^Y 3 + =0 . . . (6.). Divide this equation through by a^y* 1 , 4- . . . =0, or a, a 2 - y a which by equation (a.) becomes _ from which, by algebra, one or more constant values of x <0 y _rp can be obtained. In case r and are both positive, x to y = c becomes y to c <0 of, and is of the form y i = ax r . . . (e.). But in case one intercept, as r , is negative, fj let TO = TO ' } then x io y = c, or y to x r r = c to , 172 HIGHER ALGEBRAIC CURVES. which is of the form y t x r = a . . . (d.). We shall call the reduced equations (c.) and (d.) "the approximate equations," in distinction from (b.) " the general form of the approximate equation for any assumed line." 444. ScJtol. The approximate equations (c.) and (d.) for parallel lines can differ only in the value of the coefficient a. _ r o For, in the equation x T Q y = c, the, exponent, which is the negative ratio of the exponential intercepts, has, by similarity of triangles, the same value for all parallel lines. And conversely, all approximate equations that differ only in the value of the constant are for lines which are parallel. For, since the ratio of the exponential intercepts is the same (by similar triangles), the lines are parallel. 445. E. G. In the example, Art. 435, the general form of the ap- T t proximate equation for the assumed line - + = 1 4 4 is oV - aVy + kttff - a V + ay = 0, or x* x 3 y -\- 4% 2 y 2 which is symmetrical in x and y, .'. by algebra - = - and ." . y x y But - = 1 does not satisfy the equation, therefore all values of y y are imaginary. T t For the parallel to the assumed line, --\-- = l t the approximate o 3 equation is #aV 2a 3 xy* = 0, . , x y \/ 1. T t And for the other parallel, - + - = 1, we also obtain x = y |/ 1. o o T t For the line - -\ = 1, the equation is etc - 174 HIGHER ALGEBRAIC CURVES. And, second, since by Art. 444 all parallel lines have approxi- mate equations which differ only in the value of the coefficient a, it is therefore evident that the expression (e.) above differs from that which we should get by replacing y with some value ro cV obtained from some parallel line only in its coefficients. But, third, since all the terms along any one of the parallel lines are of the same degree in x after the replacement, all the terms along each of the lines are of the same degrees respectively as the terms situated at the intersection of each of the lines with the axis of r. But the terms along r increase uniformly in degree from the origin ; therefore each successive parallel count- ing from the origin has its terms of higher degree than the preceding. M7. Schol. This may appear more clearly if the facts be indicated upon a diagram. Let us write the literal part of every term in a xy 1 x 4 x 10 x n x s x x w x 6 x~ X s x 5 ^c x a- 7 X s x~ a 8 " > <, x X 7 1 X x~ a: 3 x^ ^L x * 2 4 6\/ general equation of the sixth degree just at the right and above its representative point. Suppose the assumed line to be - + - 1; 6 3 .'. (equation (c.)), y ax 1 , for, = 6 and t 3, and the terms thus become by the replacement of y as seen in the right hand figure. RELATIVE DEGREE OF TERMS. 175 And the same result would have appeared had we made use of any line the ratio of whose intercepts is the same. 448. Cor. When one intercept is infinite (as ), we have When TQ IQ, then y = bx. ^Proposition 7. 449. Tlieorem.In order further to compare the degrees of the several terms in the equation of any curve, replace y by its value obtained from the approximate equation for any assumed line of tine form y t x r = a (the intercept on the axis of t being negative, and that on r positive) ; then, 1st. All terms whose representative points are on the assumed line become of the same degree. 2d. All the terms whose representative points are on any one line parallel to the assumed line become of the same degree. 3d. All terms whose representative points lie between the line and the origin become of less degree than those whose representative points are on the line, but all terms whose representative points are on the opposite side of the line become of gr 'eater degree. 4th. If x be replaced instead of y, the terms whose representative points lie on the side toward the origin are of greater degree, and those on the opposite side of less degree. 5th. A corresponding statement is true when the intercept on the axis of r is negative, and that on t positive. For, the first, second and third parts of the proposition are proved in the same manner as in Art. 446 ; and fourth, since i_ ^ y t x r = a, .'. x (ay~ t ) r , and y = (ax~ r } t i.e., the greater powers of x are the less powers of y, and vice versa. 176 HIGHER ALGEBRAIC CURVES. 450. Schol. This proposition may be illustrated as was the pre- vious one. Assume the line 451. Cor. When one inter- cept is infinite (as ), we have X xl *1 I 2. Exercise. Write the literal parts of the terms in a similar manner when x is replaced. Proposition 8. 453. Theorem. An approximate equation of the form y* = ax r for a side of the exponential polygon of the first kind (Art. 439) represents a curve which approximates to the original curve for infinitesimal values of x and y (i. e., near the origin) ; and an approximate equation for a side of the second kind is of the same form, but represents a curve that approximates to the original curve for infinite values of x and y. For, if the general form of the approximate equation for a side of the first kind be obtained in the manner shown in Art. 443, equation (6.), and the value of y obtained from this equation be substituted in every term of the equation of the curve, it has been shown (Art. 446) that the terms whose representative points lie on this side are then of lower degree than the rest ; for, this side lies between all others and the origin. When x is made infini- tesimal after the replacement (i. e., both x and y infinitesimal in the original equation), all other terms become vanishing quantities in comparison with those whose representative points lie on this side. In the same way it is evident that the terms whose repre- sentative points are on a side of the second kind are of higher degree than the rest, and that all other terms are vanishing quantities, when infinite values are assumed for x and y. APPROXIMATE CURVES. 177 454. Examples. Show that we have the following approximate curves. (1.) In the Cissoid, x 3 =py' i , (2.) In the Witch, f = 2ax, x 2a. (3.) In the Cubic Trisectrix, y=.x (4.) In the Folium, y = x, x = a. (5.) In the Lemniscata, and the equilateral hyperbola, x = y. (6.) In the Limacon, y=x -\/~3, (7.) In the Cardioid, x 3 = \ ay\ Proposition 9. 455. Tlieorem.tAn approximate equation of the form for a side of the third kind represents a curve that approxi- mates to the original curve for infinite values of x and infinitesimal values of y, when its intercept on the axis of r is positive, and that on t negative (that is, for that branch of the hyperbolic approximate curve to which the axis of x is an asymptote) ; but it represents a curve that approximates to the original curve for infinite values of y and infinitesimal values of x, when its intercept on the axis of t is positive, and that on r is negative. For, the terms whose representative points lie on a side of the third kind are of higher degree in x and lower in y (or vice versa) than any other terms in the equation (Art. 449) ; there- fore, if x = co nearly, and y nearly, all other terms are vanish- ing quantities. 456. Schol. 1. The general form of the approximate equation for a side of the fourth kind contains x only, or else y only. Suppose it is x only, then the other terms of the original equation vanish when y = 0. Hence we obtain the points of intersection of the curve with the axis of x. 12 178 HIGHER ALGEBRAIC CURVES. 457. Schol. 2. The general form of the approximate equation for a side of the fifth kind contains the same powers of either x or y, according as the side is parallel either to the axis of r or t. Suppose each term has in it the same power of x, and on dividing through by that factor the equation will consist only of powers of y (i. e., y = a by Art. 451) ; then all the terms of the original equation whose repre- sentative points do not lie on this side vanish in comparison on making y=a and x = oo nearly (i.e., the curve approximates at infinity to the line y = a). A similar statement is true when x = & constant. Proposition 10. 458. Theorem. 1st. When an equation has no constant term i. e.,no representative point at (0, 0)the curve has a single branch through the origin i. e., it has a single point at (0, 0). '^2d. When an equation has no constant term nor terms of the- first degree i. e., no representative points at (0, 0), ^( 1, but not W W * necessarily an integer. (If n = ^ the curve is a parabola.) (9.) Discuss the double points of the curves in Chapter IX. 401. Exercise. Move the origin in the example, Art. 435, to the point (a, 0), and show that the transformed equation then has at (0, 0) the approximate equations x = %, and y 2 = 2ax, and at (0, oo) the approximate equation x l y = a 3 . 462. General Scholium. The properties of the exponential poly- gon may also be used to discover the equation which will represent the general contour of a given curve whose equation is unknown. CHAPTER XI. TRANSCENDENTAL CURVES. 463. Transcendental Curves are those which require the use of transcendants, such as sin, tan, log, etc., to express the relation between the x and y of any point. Several of these are discussed in the following propositions. Proposition 1. 464." Theorem. The equations x r (

) and y = r (1 cos )] r (2mz -f

m sin ^>), and y=r(lm cos (p) represent the locus of a point on the radius of a circle which rolls along a straight line ; in which mr is the dis- tance of the point fro?n the centre of the circle, and the curve is a prolate or curtate cycloid according as m 1. The curve is also called a trochoid. For, since

and y = r (1 cos }. 470. ^Proposition 3. TJieorem.The equations ( \ / r i + r 2 X ~ ( r i ~*~ r 2J COS ( f~ T 2 COS ( ^ \ 2 V ^i ~T~ T 2 j Sin ^ 7*2 SI represent cun epicycloid, which is the locus of a point on the circumference of a circle which rolls around on the outside of a fixed circle; in which TI is the radius of the fixed and r-i that of the rolling circle , and y its angle of revolution. For, let OB = r l , the radius of the fixed circle, and CB = CP = r 2 , the radius of the rolling circle, when P is the point which was originally in contact with A. Let the angle of revolution AOC= ', I A then x = ON+ NM= (r x + r 2 ) cos tp + r z sin I

r 2 sin (up +

| sin 2

) 2 + a 2 cos ^(1 cos ^) = a 2 (^ cos ^), which is the equation of the cardioid (Art. 421). 472. Exercise. Show that the equations = (r l j r r 2 ) cos mr 2 cos I V) = (n + r a ) sin p mr 2 sin (- -^ J V r / represent the prolate and curtate epicycloid i. e., the epitrochoid. ^Proposition 4. 473. TJieorem.The equations = (r l r 2 ) coa

1 represent an hypocycloid, which is the locus of a point on the circumference of a circle which rolls around the inside of a fixed circle. For, these equations may be obtained by a method similar to that used in the previous proposition, or by putting r 2 for + r 2 . 184 TRANSCENDENTAL CURVES. 474. Schol. 1. When r t = #r 2 , the equations become x = 2r.i cos , and y =-0, in which x may have any value between -f r l and r 1} and the curve is reduced to the diameter of the fixed circle. 475. Schol. 2. When ^ = ^r 2 , we have x = 3r. 2 cos

) and y r (sin ^ (p cos $p) represent the involute* of a circle, which is the locus of P at the end of a thread unwound froin the circumference of a circle whose radius is r. * This involute is a spiral which has no real points within the generating circle. REPEATING CURVES. 185 For, if then OBN^- BPD = ^-v, & and BP = AB=--ry, ... x = OM=ON+DP, and y = MP = NB-DB; . ' . x r cos

sin cos ^. 478. Exercise. Show that the polar equation of this curve is or TRIGONOMETRIC LOCI. Proposition 6. 479. Theorem. The equations y = sin x, y = tan x, y = sec x, represent periodic or repeating curves. For,, let the curves be constructed by tracing them through points determined by tables of natural sines, tangents or secants, or by drawing the lines as in Fig. 1. We then have the curve of "X Fig 2. Figl. 186 TRANSCENDENTAL CURVES. sines, Fig. 2; the curve of tangents, Fig. 3; and the curve of secants, Fig. 4. Since sin x = sin (2nn + x), and tan x = tan (2nn + x), and sec x = sec (2nx + x) (when n is an integer), the curves repeat themselves along x. Fig3. X Fig 4. X 480. Schol. 1. The equation x s x> y = ^x = x- + - etc., by trig., . * . the right line y = x approximates to the curve at the origin (Art. 453) i. e., is tangent to it. Move the origin to (^TT, Ij, .-. x = af + jji: and y =-- y' + 1, .'. tf + 1 = sin(#' + |TT) '2 yft = 'coBx f = l + etc., by trig., .'. the parabola x n = 2tf & (sin y, sin x) ; for as P moves along its locus parallel to the axis of y, P' moves along its locus parallel to the axis of x. When PS is tangent, P'S is likewise tangent. From this it follows that when the axis of x is tangent to

(x, y) 0, and PT is also the line KE or x= 1, then P' T is not tangent to

(sin y, sin x)=0 represents a curve (or calico pattern) such that one entire pattern which lies within the square whose sides are repre- sented by the equations x = ^~ and y = -' - is in- definitely repeated in each direction in such a manner as 190 TRANSCENDENTAL CURVES. would be caused by a kaleidoscope of four mirrors whose cross-section is the square mentioned. For, it is noticeable that the line PS^ intersects the curve y = sin x in an infinite number of points S l} S 2 , 8 3 , etc., on both sides of 0. X Also that PjPj intersects the curve = sin y in an infinite number of points T l} T 2 , T 3 , etc., above and below 0, and that from 8 n and T m we can obtain P f nm . Again, as P is moved along, (x, y) = is tangent to x = 1 or yl, then the correspond- ing part of v(sin y, sin x)-0 may meet a? = db'|* or y=^n at any angle. Proposition 10. 490. Theorem. If ? ( x , y) = be traced and P be any point of it, and the auxiliary curves x = tan y and y = tan x be also traced, then P' the point of

(x, y) = are used in construct- ing

(x, y) 0, and the auxili- ary curves x = sin y, 7<# y = tan #, 7^e locus y (sin y, tan x = 0) may be constructed, one entire pattern of which lies within the square xjj-, y j,-, and is repeated indefinitely without change of form between the parallels y = dz J- JT, #7^0 repetition being such as would be caused by a reflection in two plane mirrors whose cross-section is y = ^^. For, this may be made to appear by a mode of demonstration similar to that of the preceding propositions. 496. Schol. Only that part of (sin y, tan x) = which lies between the parallels y = l. Proposition' 13. 497. Theorem. By the aid of

(sec y, tan x) = 0,

, 0) = 0) represents a polar curve or spiral which may be readily traced from its relation to the original curve (viz.

= ad, by putting x for p and y for 6, 194 ANALOGOUS CURVES. 195 becomes x ay. Let a = j say. Draw the line x = \y. Also draw a circle with radius OA = 1, which is the measuring circle. The arc 6 is measured off in linear units upon the circumference of this circle. A\YX When 0'N l =y = = ^n, then When 0'N 2 =y = = fjr, then N 2 P 2 =x = p = -j,ic, etc., etc. And a similar construction may evidently be made of any polar curve from its analogous rectangular curve. 502. SchoL 1. The polar curve p aQ is the spiral of Archimedes, and it is analogous to the right line through the origin. In it the radius vector p is evidently proportional to the arc of the measuring circle. Since each value of negative as well as positive gives a single value of p, the spiral is symmetrical about the axis of y. 503. Schol. 2. The circle /> = a is analogous to the line x = a parallel to the axis of y, and the fixed line = a is analogous to the line y = a parallel to the axis of x. 504. Exercises. Construct the spirals, (1.) f-l* P + P = 0. (3.) 196 SPIRALS AND POLAR CURVES. Proposition 2. 505. TJieorem. Parabolic Spirals represented by equations of the form in which r and t are different positive integers, all have the initial line tangent to them at the pole. For, let P 2 and P 3 be upon the spiral, and let the angle QP S P 2 = (p. Then if P 2 P 3 is infinitesimal, we may consider the perpendicular QP 2 = arc p^^ O^) as it is when 2 = 6 3) . tan = . . (a.) 3 a 2 as in Art. 423, when P 2 P 3 is infinitesimal. Find the value of and then make Pi=P 2 = PS and O l = d 2 = 3 , . ' . from (a.) tan = T tan d> = i in which (p is the angle between the tangent and radius vector of (ft, 0,). Letd 1 = t then ft = 0, and tan^ = 0, . * . the tangent to the spiral at the pole has the same direction as the initial radius vector and initial line. 506. Schol. 1. When r is even and t is odd, the spiral has for every value of two numerically equal values of p with opposite signs, so that every polar chord is bisected at the pole. \ PARABOLIC AND HYPERBOLIC SPIRALS. 197 507. Schol. 2. When r is odd and t is odd, the curve is symmetric about the axis of y, for p and may be written for p and without altering the equation. AT; 508. Schol. 3. When r is odd and t is even, the \/' curve is symmetric about the axis of x, for equal A values of of opposite sign give the same value y \ of p. ) X 509. Schol. 4=. When r and t are both even, the equation may represent at least two of the previously mentioned spirals. 510. Schol. 5. The spiral whose equation is /o 2 = ad is frequently called the parabolic spiral. 511. Schol. 6. If + be put for (Art. 49) in the equation of any parabolic spiral, the curve is no longer tangent to the initial line, but is tangent instead to the line = , for the curve is rotated through the angle Q . Proposition 3. 512. TJieor em. Hyperbolic spirals represented by equations of the form in which r and t are positive integers, all approximate to the initial line at infinity, and approach the pole by an infinite number of revolutions of the generating line. ft ft For, find the value of -- 2 , as in Art. 427, and substitute it ft -A 1 198 SPIRALS AND POLAR CURVES. in (a.) of Art. 505, then make p p z = /? 3 and Oi = 2 = 6 B . .'. tan^= -fa (6.). Let O l = 0, then p l = oo, and tan = 0, . . the spiral approxi- mates to the initial line at infinity. Again, if f>0, then 6 = oo ; . * . the spiral approaches the pole as 6 becomes infinite. 513. Schol. 1. When r is even and t is odd, each polar chord is bisected at the pole. 514. Schol. 2. When r is odd and t is odd, the spiral is symmetric about the axis of y. X 515. Schol. 3. When r is odd and t is even, the spiral is symmetric about the axis of #. 516. Schol. 4. When r is even and t is even, the equation may represent two separate spirals. 517. Schol. 5. The spiral pO = a is often called the hyperbolic spiral, and is asymptotic at infinity to a line parallel to the initial line and tangent to the circle p = a. The Lituus p z = a is asymptotic to the initial line at infinity, having the point of contraflexure THE EXPONENTIAL POLYGON OF SPIRALS. 199 518. Schol. 6. If + be substituted for (Art. 49), in the equa- tion of any hyperbolic spiral, the curve no longer approximates to the initial line at infinity, but approximates instead to the line == Q . 519. The exponential polygon may be applied to finding spirals which approximate in shape and position to those parts of polar carves for which /> and 6 are infinite or infinitesimal. The same principles are applicable to the determination of the approxi- mate equations which can be derived from any given equation in p and 0, as were enunciated and proved in Chapter X. with respect to x and y, but the interpretation of the approximate equations so obtained needs consideration. We shall state the interpretation of these approximate equations with- out formal proof, as the interpretation appears at once from the analogies already pointed out between polar and rectangular equations. For convenience we shall consider that the spiral of Archimedes is one of the parabolic spirals. 520. Sides. The exponential polygon, when applied to alge- braic equations in p and 0, has sides of eight kinds, viz. : 1st. Sides whose intercepts on the axis of r and i are both positive, and which lie between the origin and the rest of the polygon. The approximate equation for such a side represents a parabolic spiral which approximates to the curve of the original equation near the pole. 2d. Sides whose intercepts on the axis of r and t are both positive, and between which and the origin lies the rest of the polygon. The approximate equation for such a side represents a parabolic spiral which approximates to the original curve for infinite values of p and 0. 3d. Sides whose intercepts Dispositive on the axis of r, and negative on the axis of t. The approximate equation for such a side represents a hyperbolic spiral which approximates to the original curve for infinite values of /, and infinitesimal values of i. e., to the initial line at infinity. 4th. Sides whose intercepts are negative on the axis of r, and positive on the axis of t. The approximate equation for such a side represents an hyperbolic spiral which approximates to the original curve for infinitesimal values of p and infinite values of i. e., to the pole. A side whose intercepts are zero falls under the third or fourth case, according as all the rest of the polygon is above that side or to the right of it. 200 SPIRALS AND POLAR CURVES. 5th. Sides that coincide with the axis of r. The approximate equation for such a side gives the value of p when = i. e., the first intercept on the initial line. 6th. Sides that coincide with the axis of t. The approximate equation for* such a side gives the value of when p = i. e., the direction of the tangent at the pole. 7th. Sides parallel to the axis of r, and having the rest of the poly- gon between themselves and the origin. The approximate equation for such a side gives a constant value of p when is infinite i. e., the circle to which the spiral is asymptotic. 8th. Sides parallel to the axis of t,- and having the rest of the poly- gon between themselves and the origin. The approximate equation for such a side gives a constant value of when p is infinite i. e., the direction of a radius vector which approxi- mates to the position of an infinite branch of the spiral. 521. Examples. Find the spirals which approximate to 7r) 2 = a(0 2 -7r 2 ). (3.) aY=(p-a)*0\ 522. Exercises. (1.) If a line OT be drawn from the pole per- pendicular to the radius vector OP = p, and intersecting at Tthe tan- gent TP, then OT=p' is the polar subtangent of P, and (Arts. 505, 512) P ' = p tan 0. If p = aO* , show that the locus of T is p' = -(0j 7:)'+', in which t is positive, negative, integral or fractional. (2.) Discuss the curves represented by the equation -- = sin - ; they will illustrate the statements of Art. 234. It will be found that when T = a there are one or more cusps at the pole, and when r < a the cusps are replaced by loops. E. G. When n = l, if r = a the curve is a cardioid, but if r = 2a it is a lima^on. If r > a, the curve does not pass through the pole. Again, it will be found that an entire branch of the curve is com- pleted by the revolution of p through 2mt. If n is commensurable, after the completion of a certain number of branches, no new branches are described by the further revolution of p. When n is a vulgar fraction in its lowest terms, the numerator states the number of revo- lutions of p before the figure is redescribed", and the denominator the number of similar branches in the complete curve. If n is incom- mensurable, both these numbers are infinite. INITIAL FINE OF 25 CENTS BOOK SS OVERDUE. S.VSNTH OAV LD 21-100m-12,'43 (8796s) '. UNIVERSITY OF CALIFORNIA LIBRARY