2 2 2 3(b√r ́ 2+b2+r22 log. √r'2+b2+b−l√r'2 + l2 —r' 3 log. (√ r2 2 + l2+1))' in which r'= 1-C s. Frustra of Spheroids, or Ellipsoids of Revolution, ec df, Fig. 104, on page 182. Distance from centre of spheroid= d (2a2-d2) a repre the height of the frustrum. senting the semi-transverse, and b the semi-conjugate axis, and d Surface of a Frustrum of a Circular Spindle, Fig. 7, p. 283. The distance from the centre of the spindle= 2(h-D.2)' and r being the radii of the two bases, e and s; h the distance between the two bases; D the distance of the centre of the spindle from the centre of the circle, as a o; z the generating arc, expressed in units of the radius. Fig. 7. a Surface of a Segment of a Circular Spindle, as b c, Fig. 8. Fig. 8. the radius of the base; the other symbols the same as given on page 282. NOTE. This last formula is essentially the same as the following. respectively and ;; g=fo, l=oc, r-radius of circle=a d, a r =ao=√r2—g2, and b=radius of end circle of segment. Paraboloid of Revolution. (See Fig. 65, on page 126.) 1 (p2+b2)3 (3 b2 — 2 p2)+2 p3 Distance from vertex: 10 p a=altitude, b=radius of base, and ·p= 2 a MENSURATION OF SOLIDS. Cycloidal Spindle. To ascertain the Contents of the Frustrum of a Cycloidal Spindle, be co, Fig. 9. 3 -17 ?' (2 r—1)* (2 1*# + 5 i* r + 151* r2 — 15 r3 ver.sin. +15 pr3)) dc =contents, I representing e f, r= and p, as before, 3.1416, 2' To ascertain the Contents of the Segment of a Cycloidal Spindle, -17 abe, Fig. 9. r2)) = com (15 r3 ver. sin. ~—- — (2 r—1)3 (2 1* +513r+ r tents. SOLIDS OF REVOLUTION. To ascertain the Volume of a Solid of Revolution,* Fig. 7. Let A X, the axis of x, be the axis of revolution, and c m d the generating curve, A n=x, m n=y, the co-ordinates of any point of the curve, and let the solid be terminated by planes perpendicular to the axis, cutting it at o and s. Let A o=a, and A s=b, the abscissæ of these points; o c =r, and s d=r′, the radii of the two bases. The origin, A, may be taken at any convenient point on A X. The general formula,† when A X is the axis of revolution, is V=pfy dx, in which p=3.1416; f is the symbol of integration, and d that of the differential. If, in the expression for V, y2 or dx be eliminated by means of the equation of the generating curve, and the integration be effected between the limits xa and x=b, or y=r and y=r', the value of V is determined. Corollary.-If A Y, or the axis of y, is the axis of revolution, then, V=pfx2dy, which differs from the preceding simply by the interchange of the letters x and y. *By Professor J. H. C. Coffin, U. S. N. This formula is thus read: The volume is equal to p times the integral of y2 multiplied by the differential of x. |