:: the whole moving force or tension of the string is (y + uta - r). Hence the accelerating force is 9 + 1 + a - 2 a F = the velocity of the chain, when quitting the quadrant, is given by V2 = {8r + 4 (2a – r)* } 4a 632. Let y, y, be the radii of the required annulus, and I the distance of its centre from the pole; also let u, b, be the semiaxis of the generating ellipse; then it may easily be shewn that Again, if u denote the radius of any of the concentric circles which compose the ring, the attraction of any particle in its circumference upon the pole is 22 +32 and by the resolution of forces the attraction in the direction of the axis 1 Hence the whole force of the circle in this direction is 2TNI (u? +12) and that of the annulus is ... udu 2πα S (u2 +2?}} taken between u = y' and u = y; that is 1 1 2πα (y2 + x2 N or 2 (26x– x?)+r? – ) = max. m Hence, and by the question, bWx (2ba a? - 62 . x) b 1 2 26 N 3 2V * (2ba' - a - b. x) (a* – 60) = 0 (865 a4)} and x = X 2ba? 2 (ba® 3'5 a) 6 which gives the position of the annulus for spheroids of all eccentricities. 633. Let the distance of the particle attracted from the centre of the sphere be a, r the radius of the sphere, and suppose any circular section whose radius is y to be made by a plane I to a, and distant from the point by the interval x; then if y' denote the radius of any o concentric with the former, the attraction of any particle in the circumference of this circle is 1 *+2) and this resolved into the direction of x is Hence the attraction of the whole circumference is 2ncy' (y^2 + x2) y'dy' ATTRACTIONS. taken between ý' = y, and y' = 0; that is 1 3 3 (yo + x)} dx (y +x*)! taken between the limits of x =a + 1, x = a - r. But by the equation to the circle ya = gole – (a - x) d.x ; and we have for the whole attraction S1 atr 634. If x be the altitude of the cone, and s its slant side, the whole attraction is (Vince, p. 142) x® = U. Again, supposing the given quantity of matter to be a3, and the density to be constant, we have 7. (s* - **) x 2 a 3 whence by substituting for x &c. &c. I may be found exactly or by approximation. 635. Let c be the distance of P from the surface of the sphere, r the radius of the sphere, then ctor is the radius of the 2 generated sphere. Let any circular section of the sphere be made by a plane I to c or to the axis, and let y be the radius of that section, and y' that of any circle concentric with it; also let x be the distance of P from the plane of this section; then the attraction of any particle in the circumference of the circle whose radius is y', is 1 y'a + rs and the attraction of the whole circumference is 2πυ y' + x89 Hence the attraction of the whole section is 2y'dy = al. (y'? + ?) + C y + taken between yʻ = 0 and y' = y; it is :: y' + x2 .cdr + 2.3 2.(c + r) 2.(c + r)x - c(c + 2r) and we finally obtain, after finding the correction on the supposi |