This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1843 Excerpt: ...perpendiculars ON, PM, pm on the axis A a; then if the equation to Pp be y = ax + (i, the equation containing the values of y at the points P, p will be 262/3 6'?(/32-a2a2) Now, since in any quadratic equation the coefficient of the second term, with its proper sign, is equal to the sum of the roots with their signs ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1843 Excerpt: ...perpendiculars ON, PM, pm on the axis A a; then if the equation to Pp be y = ax + (i, the equation containing the values of y at the points P, p will be 262/3 6'?(/32-a2a2) Now, since in any quadratic equation the coefficient of the second term, with its proper sign, is equal to the sum of the roots with their signs changed, 2 b2 8 = PM + pin. To obtain the relation between X and Y, we must eliminate 8 between (l) and (2); aV + 6 a2 a2 + b b2 a2 a, Y--U Now, since (a) remains the same for all chords parallel to Pp, (25), the equation just found expresses the relation between the co-ordinates of their middle points; and being of the first degree, the locus required is a straight line. Def. The straight line, which has been proved to be the locus of the middle points of any number of parallel chords, is called a diameter, and the points in which it intersects the curve are called the vertices. b2 163. Coh. The equation Y = X, is the equation a. a to a straight line passing through the origin, which in this case is the centre; hence every diameter must pass through the centre. 164. A diameter being drawn through a given point, to find the equation to any one of its ordinates. If x', y be the co-ordinates of the given point, the equation to the diameter drawn through it will be y =--x 0) x Let y = ax + ($ (2), be the required equation to any ordinate, v 6a then--, ----(162); x a2a - y hence, any ordinate to a diameter passing through (x, y) has for its equation 62 ' a y =---'--x + P a y 165. Coit. Comparing this equation with the equation to the tangent (Art. 140), it appears, that the tangent applied at the vertex of any diameter is parallel to the ordinates of that diameter. 166. Any two diameters being given, if the ordinates of one be parallel to the ot..
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