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angle arbitrary axes axis becomes centre centre of gravity circle circumscribed coefficients collinear common conic section conjugate consequently contained coordinates corresponding curve denote described determine diagonals diameters direction distance divided draw drawn eight ellipse ellipsoid equal equation evidently expression figure forces four function give given plane given points given surface hence inscribed integral intersection joining known locus manner meet multiplied obtain opposite origin parallel pass perpendicular plane polygon positive problem produced properties proposed quadrilateral radius ratio reduced reference relation represented respectively result roots second degree second order sectors sides similar similarly SOLUTION sphere square straight line substituting suppose surface taken tangent theorem third touch triangle values vertical
Page 225 - The line which joins the mid-points of two sides of a triangle is parallel to the third side and equal to one half of it.
Page 340 - The sum of the squares of the reciprocals of the areas of the sections of an ellipsoid made by any three diametral planes at right angles to each other is constant.
Page 283 - V ; shew that if 0 be the limiting value of the angle of inclination of the beam to the horizon at which it will rest, W its weight, and R, R
Page 218 - A merchant bought a cask of wine for |48, and sold a quantity exceeding three-fourths of the whole by two gallons, at a profit of 25 per cent. He afterwards sold the remainder, at such a price, as to clear 60 per cent, by the whole transaction, but if he had sold the whole quantity at the latter price, he would have gained 175 per cent. How many gallons were in the cask ? — Ib.
Page 335 - ... join DB. Then AD is equal to the given straight line. And the angle ACB is equal to the sum of the angles CDB and CBD (I. 32), that is, to twice the angle CDB (I. 5). Therefore the angle ADB is half of the angle in the given segment. Hence we have the following synthetical solution. Describe on AB a segment of a circle containing an angle equal to half the angle in the given segment. With A as centre, and a radius equal to the given straight line, describe a circle. Join A with a point of intersection...
Page 192 - If straight lines be drawn from the angles of a triangle through any point, either within or without the triangle, to meet the sides, and the lines joining these points of intersection and the sides of the triangle be produced to meet ; the three points of concourse will be in the same straight line.
Page 171 - A merchant bought a cask of spirits for £48, and sold a quantity exceeding three-fourths of the whole by two gallons at a profit of £25 per cent. He afterwards sold the remainder at such a price as to clear £60 per cent by the whole transaction...
Page 23 - L e** + 1 where the upper or lower sign is to be taken according as...