Publisher's Synopsis
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. 1870 Excerpt: ...is, the volume is equal to the surface by one-third the radius. 2. To find the volume of a spheroid of revolution. There are two species of spheroids of revolution. 1st. The prolate spheroid, generated by revolving an ellipse about its transverse axis. 2dly. The oblate spheroid, generated by revolving an ellipse about its conjugate axis. 1st. The prolate spheroid. In this case the equation of I2 the meridian curve is, y2 =--(a2--x2). Substituting a this in (1), and integrating between the limits x =--a, and x = + a, we have, + a V" = 7rjrf(a-x2)dx = b2a = r& X 2a (3) That is, the volume is equal to two-thirds of the circumscribing cylinder. 2dly. The oblate spheroid. In this case, if the conjugate axis coincide with the axis of x, the equation of the meri a2 dian curve is, y2 =--(b2--x2). Substituting in (1), and integrating from--b to + b, we have, V" = -a2b = %ra X 26 (4) o o Hence, as before, we have, the volume equal to two-thirds the circumscribing cylinder. In both cases, if a--b--r, we have, F" = r3 (5) This hypothesis causes the ellipsoids to merge into the sphere. 3. To find the volume of a 'paraboloid of revolution. The equation of the meridian curve is, y2 = 2px. Hence, from (1), we have, V = 2pjxdx = -px2 + C (6) If the initial plane pass through the vertex, we have (7=0, and Vf = -px2 = -y2 X x.... (7) That is, the volume is equal to half the cylinder that has the same base and the same altitude. 4. To find the volume generated by revolving one branch of the cycloid about its base. The differential equation of the meridian curve is, dfc.-ydy hence, from (1), we have, ysdy /2ry--y2 Reducing by Formula E, integrating the last term by Formula (26), and taking the integral between the limits, y = 0, and y = 2r...
