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angle angular 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 geometrical given plane given points given surface gives hence inscribed integral intersection joining known locus manner meet Multiply 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 suppose surface Table taken tangent theorem third touch triangle values vertical
Pàgina 67 - B' sin B sin C' sin C sin A cos a = cos 6 cos c + sin b sin c cos A. with similar formulae for cos 6 and cos c. cos A = — cos B cos C + sin B sin C.
Pàgina 330 - 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.
Pàgina 327 - 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
Pàgina 208 - 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, but if he had sold the whole quantity at the latter price, he would have gained 176 per cent.
Pàgina 325 - ... 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...
Pàgina 13 - F(z) given by (2) we obtain _^,1_=РтГ 1 1 о«, 04 " where the upper or lower sign is to be taken according as the...
Pàgina 272 - This point is called the centre of vibration, and it is obvious that the time of vibration of a compound pendulum is the same as that of a simple pendulum whose length is equal to the distance of the centre of vibration from the point of suspension.
Pàgina 91 - t' is the distance between the points (x¡, y,, z¡) and (x, y, z) if a, b, c, are dc's of the line. Vector equation of a lin« passing through the point A(o) and parallel to vector 6 is r '= a + К b for scalar X.
Pàgina 65 - ... 0. It is therefore evident that the Archimedean axiom is not satisfied by this system of lunar angles. As for the addition of such angles, we may define the sum of the angles formed by two tangent circles as the angle formed by that tangent circle the reciprocal of whose radius is equal to the sum of the reciprocals of the radii of the two given circles. If then we have any circle through...