centres of gravity; . the centre of gravity of the pyramid is in Ag; for the like reason it is also in Bg'; it is .. in their point of intersection G, or it is in the middle point of the line joining the middle points of any two edges not meeting, of the pyramid. 265. Upon the diameter of the circle which passes through (S) let falls from A, B, C, &c., and let their distances from the common centre of gravity (G) be denoted by a, b, c, &c. Then joining AS, AG, we have SA2 = AG2 + R2 + 2Ra and A x SA2 = A. AG2 + A. R2 + 2R × ɑA, Similarly, B x SB CX SC2 B. BG'+ B.R + 2R x b. B C x CG + C. R+ 2R x c.C &c. &c. .. A × SA + B x SB2 + C x SC2 + &c., ≈ A × AG3 + B x BG + C x CG2 + &c. 1 266. The equation to a spherical surface, the co-ordinates originating at the centre, is x2 + y2 + z2 = r2. Hence the distance (Y) of the centre of gravity from the plane of (y, z,) (See Whewell's Mech. p. 101,) is Y = Sfzxdxdy limits of x = 0, and x = √2-y?, and, by the question, fzdxdy Now the 3 T. 8 sphere) being symmetrical with respect to the three co-ordinate planes, it is evident its centre of gravity must be equally distant from each of them. Hence may be determined the actual position of the centre of gravity of the proposed solid, which is the intersection of the three planes drawn parallel to the co-ordinate planes of (xy), (yz), (xz), and distant from them by the same interval T. Its distance from the centre of the sphere is 3 8 On this subject the reader will find much useful and elegant matter in the work above cited. In page 91, lines 7, 8, two errors of calculation having fallen under our observation, we notice them, with the view of facilitating the perusal. we have the distance of the centre of gravity of its arc s from the axis of y originating in the vertex expressed by -ᄒ. dx √ er x √ √ 2r = x. dx + √ er faz vers. " x 2√2r. x . (2r−x)o + 2r √↓ 2r − x + √x: vers.~x— X and Y determining the position of the centre of gravity (G) of an arc (s) of the cycloid measured from the vertex wholly on one side of the axis of x. If in like manner X', Y', be found determining the centre of gravity (G) of an arc (s) on the other side of the axis, their common centre of gravity, or the centre of gravity of the arc s+ $ may be found by dividing the line joining G, G', in the proportion of g' to g. Hence it appears the centre of gravity of any portion of the arc of a cycloid may be found. If ss', the centre of gravity will be in the axis distant from the vertex by 268. x 3 This problem is reducible to "If straight lines PQ (Fig. 82,) be made to cut off from two sides AC, BC, of a ▲, segments PA, QB, whose sum = a constant quantity (m) and PQ be divided in G so that PG GQ :: BC: AC then the locus of G is a straight line parallel to the base AB; which admits the following proof: Let fall Pp, Gg, Qq, LAB, and produce QP, BA, to meet in R. Then from similar A, we get Qq: Pp :: PQ + PR : PR ..QqPp Pp :: PQ: PR, similarly and PG PQ :: sin. A sin. A + sin. B. Hence, by substituting, we get ..the locus of G is a straight line parallel to AB. If the question be considered mechanically; then since P and Q are in stable equilibrium in every position, their centre of gravity is the lowest point possible. Hence whatever may be the position of P, Q, their centre of gravity G is at the same distance from the horizontal base AB; i. e., the locus of the centre of gravity is a straight line parallel to the horizon. 269. The spherical sector may be divided into a cone, whose vertex is the centre of the sphere, and a spherical segment, having the same circular base. Let a, a', be the altitudes of the cone and spherical segment, r the radius of the sphere. Then (p. 199.) the distance (g) of the centre of gravity of the cone from the centre of its base is and the distance (g) of the spherical segment from the same But a 7-a; . by substitution and reduction, we have Again, V, V' denoting the volumes of the cone and segment, we have .. calling G, G' the distances of their common centre of gravity .. the distance (D) of the centre of gravity of the spherical 3 sector from the centre of the sphere, D = a + G, is known. 4 Let a 0, or the sector be a hemisphere, then |