Page Art. 46. The equation to a line, making a given angle with a given line, is 48. If p be the perpendicular from a given point (xı yı) on the line y=x+b, 49. The length of the straight line drawn from a given point, and making a given angle with a given straight line, is 50. The perpendiculars from the angles of a triangle on the opposite sides meet in one point 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 29 30 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 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, 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 6. 35 ⚫ 35 - a) (y b) cos. w. 35 63. If the original axes be rectangular, and the pole at the origin, 64, 65. Let a and b be the co-ordinates of the centre, and 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 x' y' is 71. To find the intersection of a straight line and a circle. A straight line cannot cut a line of the second order in more than two points 40 40 Art. 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. -- 2 {b sin. + a cos. CHAPTER VI. DISCUSSION OF THE GENERAL EQUATION OF THE SECOND ORDER. 75. The Locus of the equation a y3 + bxy+cx2+dy+ex+f=0, depends on the value of 6o 76. b2 1 4ac. 4 ac negative; the Locus is an Ellipse, a point, or is imaginary, according as the roots x and x of the equation (6o 4a c) x2 + 2 (bd — 2a e)x+ d2 — 4 a ƒ = 0 are real and unequal, real and equal, or imaginary.— Examples 77. 62-4 ac 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 real and equal. Examples 78. 62 4ac0; the Locus is a Parabola when bd2ae is real; but if · 2a e = 0, the locus consists of two parallel straight lines, or of one straight line, or is imaginary, according as d2 - 4 a ƒ is positive, nothing, or negative 43 46 CHAPTER VII. REDUCTION OF THE GENERAL EQUATION OF THE SECOND ORDER. 80. Reduction of the equation to the form ay's + bxy' + c x2+f' = o. 50 81. General notion of a centre of a curve. The ellipse and hyperbola have a centre, whose co-ordinates are m= 2ae- bd n= 2cd-be 51 82. Disappearance of the term xy by a transformation of the axes through an angle 6, determined by 84. The reduced equation is a' y" 2 + c x′′ 2 +ƒ' = 0, where 52 +f 85. Corresponding changes in the situation of the figure 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 6, where tan. 20 a-e 93. The coefficient of x2 or y2 disappears CHAPTER VIII. THE ELLIPSE. 100. The equation to the Ellipse referred to the centre and axes is a2 y2 + b2x2 = a2 b2 101, 102. Symmetry of the curve with regard to its axes 103. The sq. on MP: the rectangle A M, MA':: sq. on BC: sq. on AC. 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 69 106-108. The Focus; Eccentricity; Ellipticity: 109. SP = a + ex, HP 110. To find the locus of a point P, the sum of whose distances from two fixed 111. The equation to the tangent is a2 y y' + b2 x x' = a2 b2 113. The equation to the tangent when the curve is referred to another origin 114. The rectangle CT, CM the square on AC; consequently C T is the same 115. The rectangle C M, MT = the rectangle A M, M A' 118. The directrix.-The distances of any point from the focus and from the direc trix are in the constant ratio of e: 1 119. The length of the perpendicular from the focus on the tangent, 120. The locus of y or z is the circle on the axis major 121. The tangent makes equal angles with the focal distances, 75 75 76 Art. 122. The length of the perpendicular from the centre on the tangent, 125. If CE is drawn parallel to the tangent, meeting HP in E, then PE AC 126. The equation to the normal 130. All the diameters of the ellipse pass through the centre; y = a x + c, a2a y + b2x : 0, are the chord and corresponding diameter 131. There is an infinite number of pairs of conjugate diameters; 133. Equation to the curve referred to any conjugate diameters, Page 78 78 79 136. The sq. on QV: the rectangle P V, VP' :: sq. on CD: sq. on CP 138. The ellipse being referred to its axes, the tangent is parallel to the conjugate diameter: the two equations are, a2 y y' + b2 x x'=a2b2, the tangent, a2 y y′+ b2 xx′ = 0, the parallel conjugate 139. The square upon CD the rectangle SP, HP 140. The perpendicular from the centre on the tangent, 85 141, 142. The product of the tangents of the angles which a pair of supplemental chords makes with the axis major is constant, b2 a2 143. The tangent of the angle between two supplemental chords, 145. The equation to the ellipse, referred to its equal conjugate diameters, is |