Art. 19. Through a given point M equidistant from two perpendicular straight lines, to draw a straight line of given length: various solutions 20. Through the same point to draw a line so that the sum of the squares upon the two portions of it shall be equal to a given square 21. To find a triangle such that its three sides and perpendicular on the base are in a continued progression Page 13 CHAPTER III. THE POINT AND STRAIGHT LINE. 22. Example of an Indeterminate Problem leading to an equation between two quantities x and y. Definition of a locus 23. Division of equations into Algebraical and Transcendental 24. Some equations do not admit of loci 25. The position of a point in a plane determined. Equations to a point, x=a, y = b; or (y—b)2 + (x−a)2 = 0 26. Consideration of the negative sign as applied to the position of points in 30. The distance between two points referred to oblique axes D2 = (a − a′)2 + (b − b′ )2 + 2 (a — a′) (b — b′) cos. w - 31, 33. The locus of the equation yax+b proved to be a straight line 36. Examples of loci corresponding to equations of the first order 37, 39. Exceptions and general remarks 40. The equation to a straight line passing through a given point is y — y1 = u(x-x1) 41. The equation to a straight line through two given points is 42. To find the equation to a straight line through a given point, and bisecting a 43. If y=ax+b be a given straight line, the straight line parallel to it is 25 44. The co-ordinates of the intersection of two given lines y=x+b, and y= ax + b', are If a third line, whose equation is y = a"x+b", passes through the point of intersection, then (a lab)-(x b'' — a" b) + (a' b" — a" b') = 0. 45. If and are the angles which two lines make with the axis of æ, 26 Art. 43. The equation to a line, making a given angle with a given line, is 47. If two lines y=ax+b and y=x+b' are perpendicular to each other, we have 1aa0, or the lines are 1 yax+band y == x + b α Page 27 48. If p be the perpendicular from a given point (xı y1) on the line y=zx+b, then 49. The length of the straight line drawn from a given point, and making a given angle with a given straight line, is √1+62 50. The perpendiculars from the angles of a triangle on the opposite sides meet in 51. If the straight line be referred to oblique axes, its equation is The tangent of the angle between two given straight lines is The equation to a straight line making a given angle with a given line is The length of the perpendicular from a given point on a given line is 52. If upon the sides of a triangle, as diagonals, parallelograms be described, having their sides parallel to two given lines, the other diagonals of the parallelograms will intersect each other in the same point 31 31 CHAPTER IV. THE TRANSFORMATION OF CO-ORDINATES. 53. The object of the transformation of co-ordinates 54. If the origin be changed, and the direction of co-ordinates remain the same, y = b+Y, x = a + X where x and y are the original co-ordinates, X and Y the new ones 55. If the axes be changed from oblique to others also oblique, 30 Art. 57. If the original axes be rectangular, and the new oblique, 60. To transform an equation between co-ordinates x and y, into another between polar co-ordinates r and e. y2 = (x − a)2 + (y − b)2 + 2 (x a) (y-b) cos. w. 63. If the original axes be rectangular, and the pole at the origin, 35 35 35 64, 65. Let a and b be the co-ordinates of the centre, and r the radius, then the equation to the circle referred to rectangular axes is generally If the origin is at the extremity of that diameter which is the axis of x, 66, 67. Examples of Equations referring to Circles 68. Exceptions, when the Locus is a point or imaginary 69. The equation to the straight line touching the circle at a point ay is y y' + x x = r2 or, generally, (y — b) (y' — b) + (x − a) (x − a) = 70. The tangent parallel to a given line, y = ax + b, is y = ax±r√ 1 + a2 = p2 71. To find the intersection of a straight line and a circle. cut a line of the second order in more than two points A straight line cannot 40 Art. Page 72. If the axes are oblique, the equation to the circle is (y—b)2 + (x − a)2 + 2 (y − b) (x − a) cos. w = r2. Examples. The equation to the tangent 73, 74. The Polar equation between u and ◊ is 40 ·2 {b sin. + a cos. 0} u + a2 +b2 = r2 = 0. . 41 42 CHAPTER VI. DISCUSSION OF THE GENERAL EQUATION OF THE SECOND ORDER. 75. The Locus of the equation ay + bxy + c x2 + dy + ex+f= 0, depends on the value of 62 -4ac. 76. 62 4 ac negative; the Locus is an Ellipse, a point, or is imaginary, according as the roots and r of the equation (b2 - 4 a c) x2 + 2 (bd — 2 a e) x + 4af0 are real and unequal, real and equal, or imaginary.— 77. b2-4ac positive; the Locus is an Hyperbola if x and x are real and unequal, or are imaginary; but consists of two straight lines if X1 and X2 are real and equal. Examples 4ac0; the Locus is a Parabola when bd - 2 ae is real; but if bd2ae 0, the locus consists of two parallel straight lines, or of one straight line, or is imaginary, according as d2 - 4 af is positive, nothing, or negative 79. Recapitulation of results CHAPTER VII. REDUCTION OF THE GENERAL EQUATION OF THE SECOND order. 2 80. Reduction of the equation to the form a y2 + bx'y' + cx12+f' = o. 81. General notion of a centre of a curve. The ellipse and hyperbola have a centre, whose co-ordinates are 82. Disappearance of the term xy by a transformation of the axes through an 84. The reduced equation is a' y'' 2 + c' x′′ 2 + f' = 0, where Art. 86. Definition of the axes 87, 88. The preceding articles when referred to oblique axes 89, 90. Examples of Reduction 91. Reduction of the general equation when belonging to a Parabola 92. Transferring the axes through an angle 4, where tan. 26= 93. The coefficient of x2 or y2 disappears 94. Transferring the origin reduces the equation to one of the forms, 95. Corresponding changes in the situation of the figure CHAPTER VIII. THE ELLIPSE. 100. The equation to the Ellipse referred to the centre and axes is 103. The sq. on MP: the rectangle A M, M A':: sq. on BC: sq. on A C 104. The ordinate of the Ellipse has to the ordinate of the circumscribing circle the constant ratio of the axis minor to the axis major 105. A third proportional to the axis major and minor is called the Latus Rectum 106-108. The Focus; Eccentricity; Ellipticity: The rectangle AS, SA' 66 67 68 68 69 109. SP = a + ex, HP = a − ex; SP + HP = AA' 110. To find the locus of a point P, the sum of whose distances from two fixed points is constant 111. The equation to the tangent is a2 y y' + b2 x x1 = a2 b2 113. The equation to the tangent when the curve is referred to another origin 114. The rectangle CT, CM the square on A C; consequently CT is the same for the ellipse and circumscribing circle 115. The rectangle C M, MT = the rectangle A M, MA' 116. The tangents at the two extremities of a diameter are parallel 118. The directrix.-The distances of any point from the focus and from the directrix are in the constant ratio of e: 1 119. The length of the perpendicular from the focus on the tangent, The rectangle Sy, Hz= the square on BC 120. The locus of y or is the circle on the axis major 121. The tangent makes equal angles with the focal distances, |