Spons' dictionary of engineering, civil, mechanical, military, and naval Volume 5

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. 1874 Excerpt: ...are to each other as these bases, or as the sections made by the plane containing the centres of gravity of the partial tetrahedrons. Therefore the centre of gravity of a frustumof the pyramid is upon the straight line which joins the centres of gravity of the tiro liases, and it div;des this line in the proportion exprensed by the formula A relatice to the tetrahedron, the letters B and 6 denoting in this case the bases of the frustum of the pyramid. This proposition extends to the frustum of the cone; and the bases of B and b being in this case proportional to the squares of their Radii R and r, we have, still denoting l,y x and y the segments determined by the centre of gravity sought upon the straight line which joins the centres of x r'-f-3 R' + 2 R r gravity of the two bases,-= 3,J,, + 2Itr The Sphere.--The centre of gravity of a sphere is its centre of shape. A Spherical Sector.--We may conceive the sector divided up into elementary pyramids, all of them having their summits in the centre of the sphere. The centre of gravity of each of them will be upon the radius drawn to the centre of gravity of the element of spherical surface which serves as its base at a distance of j of this radius from the centre. Suppose an auxiliary spherical surface described, with a radius equal to $ that of the sphere, and terminated in the cone which limits the sector, which spherical surface will be similar to that which forms the base of the sector. This auxiliary surface will cut all the pyramids, and the section obtained in each of them will have its centre of gravity at the same point as the pyramid. Hence it follows (Principle V) that the centre of gravity of the sector is the same as that of the auxiliary surface, and that consequently it is in the middle of the a...