a ari p== Art. Page 46. The equation to a line, making a given angle with a given line, is B . 27 Itaß 47. If two lines y = ax + b and y =á x + b'are perpendicular to each other, we have lta = , or the lines are 1 27 48. If p be the perpendicular from a given point (x1 yı) on the line y = x +b, then - 6 28 vi + 49. The length of the straight line drawn from a given point, and making a given angle with a given straight line, is yi b 29 wit B 50. The perpendiculars from the angles of a triangle on the opposite sides meet in one point 29 51. If the straight line be referred to oblique axes, its equation is sin. 6 30 ) The tangent of the angle between two given straight lines is tan. (0 - 0) = (u – a') sin. w 1 + ad + (a + a) cos. W The equation to a straight line making a given angle with a given line is u sin. w B(1 + a cos. w) Y - Y1 = (x -x1). sin. w + B (a + cos.w) The length of the perpendicular from a given point on a given line is (y1 – AXI 8) sin. w p== • 31 N{1+2 a cos.wta?}' 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 . 31 sin. (w CHAPTER IV. THE TRANSFORMATION OF CO-ORDINATES. 32 53. The object of the transformation of co-ordinates y=b+ Y, x = a t X 1 sin. w { X sin. X Ax +Y sin. Y Ar} sin. « A y = X sin.(w—0)+Y sin. (w-ad') sin. w 34 Art. Page 57. If the original axes be rectangular, and the new oblique, y=X sin. 8 + Y sin. e', 34 58. Let both systems be rectangular, y= X sin. 0 + Y cos. 8 = X cos. X Ay + Y cos. Y A y, 34 60. To transform an equation between co-ordinates x and y, into another between polar co-ordinates r and 0. q sin.(0 + ) y=b+ sin. w 35 61. Įf the original axes be rectangular, y = b + q sin. e, x = a + cos. 6. 62. To express r and 6 in terms of x and y, (y – b) sin.w 0 + 0 = tan. -1 3 – a + (y-6) cos.w p= (x – a)2 + (y – 5)2 + 2 (z a) (y – b) cos. e. 63. If the original axes be rectangular, and the pole at the origin, ข y2 + x2 g? = x2 + y2. 35 0 =tan.-1. y = sin-1 =cos. - 1 dy2+x2 36 CHAPTER V. ON THE CIRCLE, 37 . 64, 65. Let a and 6 be the co-ordinates of the centre, and r the radius, then the equation to the circle referred to rectangular axes is generally (y – b)2 + (x – a)2 = pi. If the origin is at the centre, y2 + x2 = 72. If the origin is at the extremity of that diameter which is the axis of x, y2 = 2 rx 22. 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 ac y' is y y + x2 = r2 or, generally, (y-6)(y' – 6) + (x – a) (x' - a) = 42 70. The tangent parallel to a given line, y = ax +b, is y = ax Ern1 + 62 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 38 39 Page 40 Art. (y – 5)% + (x – a)2 + 2 (y -b) (x – a) cos.w = r2. 2% – 2 cu cos. (6 - «) + 02 - 5% = 0, 41 CHAPTER VI. DISCUSSION OF THE GENERAL EQUATION OF THE SECOND ORDER. 42 . 43 75. The Locus of the equation a ys + bxy + 6x + dy + es +f=0, depends on the value of 18 4ac. 76. 62 - 4ac negative; the Locus is an Ellipse, a point, or is imaginary, according as the roots xi and x, of the equation (62 - 4ac) x2 + 2 (bd - 2a e)x+ d 4 a f = 0 are real and unequal, real and equal, or imaginary. Examples 77. 1% – 4ac positive; the Locus is an Hyperbola if Xi and xq are real and unequal, or are imaginary; but consists of two straight lines if xı and x, are real and equal. Examples 78. 62 4ac=0; the Locus is a Parabola when bd 2 ae is real; but if bd - 2ae = 0, the locus consists of two parallel straight lines, or of one straight line, or is imaginary, according as d? — 4 a f is positive, nothing, or negative 79. Recapitulation of results 46 48 49 CHAPTER VII. REDUCTION OF THE GENERAL EQUATION OF THE SECOND ORDER, 2 80. Reduction of the equation to the form ays + bactyl + call + fl = 0 . 50 81. General notion of a centre of a curve. The ellipse and hyperbola have a centre, whose co-ordinates are 2a e-bd 2cd - be 62 - 4ac' 62-4 ac' 82. Disappearance of the term xy by a transformation of the axes through an angle 6, determined by tan, 20 = 52 m n= 51 arC 84. The reduced equation is a yll2 + d " + f'= 0, where . 61 Art. a'y"? + e'x" = 0, or c'x" ? + d' yo'l 61 = 0. 62 . . . CHAPTER VIII. THE ELLIPSE. . 68 . . . . 100. The equation to the Ellipse referred to the centre and axes is ao vo + b^r°= ao so 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 Rectum69 106-108. The Focus ; Eccentricity; Ellipticity : 69 The rectangle AS, SA' = sq. on BC 109. SP = ut ex, HP-a- ex ; SP + HP = A A 70 110. To find the locus of a point P, the sum of whose distances from two fixed points is constant 70 111. The equation to the tangent is a' y y' + b2 x x' = a be 71 113. The equation to the tangent when the curve is referred to another origin 72 114. The rectangle CT, CM = the square on AC; consequently C T is the same for the ellipse and circumscribing circle 73 115. The rectangle C M, MT= the rectangle A M, MA' 73 116. The tangents at the two extremities of a diameter are parallel 117. The equation to the tangent at the extremity of the Latus Rectum is y = a + ex 74 118. The directrix.—The distances of any point from the focus and from the directrix are in the constant ratio of e:1 74 119. The length of the perpendicular from the focus on the tangent, a text e al 62 76 cy Art. Page 122. The length of the perpendicular from the centre on the tangent, ab P= Nog 125. If C E is drawn parallel to the tangent, meeting H P in E, then PE = AC 78 126. The equation to the normal as y 78 62 b 127. CG= ex'; CG' = y'; MG= X'; PG = 62 ao y – y=(2) = . 79 79 80 81 82 82 83 . The rectangle PG, PG' = the rectangle SP, HP 130. All the diameters of the ellipse pass through the centre; y = ax to c, a?a y + bax = 0, are the chord and corresponding diameter 131. There is an infinite number of pairs of conjugate diameters; 6% tan. & tan. & a2 133. Equation to the cạrve referred to any conjugate diameters, aja ya + b12 x? = a, b,? 134. a 2 + b 2 =a2 + 62 135. a, b, sin. ( 1) = ab 136. The sq. on QV : the rectangle P V, VP' :: sq. on CD : sq. on CP 137. The ellipse being referred to conjugate axes, the equation to the tangent is un' yy' + 6,9 x x' = a;% 6,2 138. The ellipse being referred to its axes, the tangent is parallel to the conjugate diameter : the two equations are, ayy' + 62 x x' = albo, the tangent, a? yy' + 6*xx' = 0, the parallel conjugate 139. The square upon CD = the rectangle SP, HP 140. The perpendicular from the centre on the tangent, a b a? 62 ; or po al +62-22 141, 142. The product of the tangents of the angles which a pair of supplemental chords makes with the axis major is constant, 62 a a 84 . 84 85 PF= dre 85 85 143. The tangent of the angle between two supplemental chords, 262 X'y — yʻx 2 62 tan. PQP' = tan. ARA'S a2–52 y2—42' al-62 144. Supplemental chords are parallel to conjugate diameters 145. The equation to the ellipse, referred to its equal conjugate diameters, is ye + 2? = ajo 146. The general polar equation, a? (y' t u sin. 6)2 +62 (x + u cos. 6)2 = al be al (1 e2) 147. The centre, the pole, u? e* (cos. 6) |