26 rotating round AB, ABD, ABC, and ABDC will generate a cone, a hemisphere, and a cylinder respectively. Now draw two parallel planes EFGH and F/F G H very near each other and perpendicular to AB, and draw FF and GG parallel to AB, then, by 80, volume generated by EHH E = irEH 2 x EE , EGG E =irEG 2 xEE , ,, ,, EFF E =irEF 2 xEE . Thus volume generated by EFF E + volume generated by EGG E = ir(EF 2 + EG 2 ) x EE = ir(EA 2 + EG 2 ) x EE = T(AG 2 ) x EE = irEH 2 x EE = volume generated by EHH E . Therefore in the limit, when the number of slices is indefinitely in creased, and their thickness indefinitely diminished, we have volume of cone generated by AF + volume of spherical zone generated by CG = volume of cylinder generated by CH. Let r = radius of sphere, h = AE = height of zone ACGE, then volume of cone = $irh- x h= gir/t 3 , and volume of cylinder = irr 2 x h, therefore volume of spherical zone = irr-h - ^irh 3 The height of a hemisphere is r, therefore volume of hemisphere = ^irr (3r 2 - r 2 ) firr 3 , and volume of whole sphere = fir?- 3 , a result readily obtainable by the infinitesimal calculus, or by inscribing within the sphere a series of triangular pyramids whose vertices all meet at the centre of the sphere, and the angles of whose bases all rest on the surface. In the limit the altitude of each pyramid becomes the radius of the sphere, and the sum of the bases of the pyramids the surface of the sphere ; hence volume =^Sxr=x 4irr 2 x r = $irr 3 . The volume of the circumscribing cylinder = nr 2 x 2r= 2irr 3 , therefore volume of sphere = | volume of circumscribing cylinder. 92. Let S denote the surface of a sphere and V its volume, then from 87 and 91 we have (a) r 7 2N/1 3/3~ .-,,==: A/T-X VV; V 47T formulae which give the radius in terms of the surface or volume, the surface in terms of the volume, and the volume in terms of the surface. 93. Volume of a Spherical Shell. Let r and r denote the radii of the two spheres, then volume of shell = V = $irr - irr 3 Now let r l -r=h, then If h be small compared with r lt then rr^ is very nearly equal to 1, and we have approximately Again, if h be nearly equal to r lt r is very small, and rjr^ is also very small, so that we have approximately 94. Volume of a Spherical Segment. Let CRC (fig. 54) be a section of a spherical segment whose altitude RQ is p, then, if OQ = 7i, J*_U volume of segment CRC = volume of Q, hemisphere - volume of zone AA C C / = |irr 3 -^ir7t{3r 2 -A 2 } , 91. B / = *^ 3 - **(r -p) (3r 2 - (r -p) 2 } = irp- (3r -p) . If we put p = 2r, we obtain as before volume of sphere = irr 3 . Again if CQ = ff.j we have CQ 2 = aJ = RQ. R Q = p(2r-p), whence r = ^ I - + ^ r 2p therefore volume of segment = JH-JJ (3aJ +p~) . 95. Volume of a S2)herical Frustum. When one of the termi- nating planes passes through the centre we have already found that the volume = frh(r*-tf), where h is its altitude. Now suppose that neither of the terminating planes passes through the centre ; for example, to find the volume of the frustum BB C C. Let RQ=p and RP = <7, then BB C C- segment RBB - segment RCC = ^q(3al + q-)- k*p(Sal+p>), where a^ and 2 are the radii of the ends CC and BB . Let q - p = h = height of frustum, and, since, from the geometry of the figure, P 1 we have volume = -nhZ(a + &D + h-}, a result which may also be obtained by considering BB C C as the difference of the two zones AA C C and AA B B. D. Spheroid. 96. Surface of a Prolate Spheroid. The prolate spheroid is the solid generated by the revolution of an ellipse about its major axis. If S be the surface generated by an arc of the curve, then taken between proper limits. In the case before us -sin- 1 ^, where c is the eccentricity (INFINITESIMAL CALCULUS, art. 179). 97. Surface of an Oblate Spheroid. The oblate spheroid is the solid generated by the revolution of an ellipse about its minor axis (fig. 55). 7 2 ^ i (INFINITESIMAL CALCULUS, Here surface = lira- + IT log, c l-e art. 179). 98. Volume of a Spheroid. We have volume of prolate spheroid /"~t~ fl / 3*~ 7T/ P ( 1 - - ) J-a V n- J - dx = 27 3?~ - ^ }dx = Similarly volume of oblate spheroid = |ir 2 6 . Thus, volume of prolate spheroid *irab 2 volume of oblate spheroid -n-a-b sphere described on major axis_ ^TI prolate spheroid sphere described on minor axis oblate spheroid Volume of a Segment of a Spheroid. (y) rb 3 ra-6 b- b 2 a- (a) The prolate spheroid. This segment is generated by the revolution of AMP (fig. 23, p. 20) about AM, and hence its volume = TT/ y"dx = IT / (Ictx - x-)dx = -- x 5- (3 - h), Jb a ~Jo where A is the origin and AM = 7. (0) The oblate spheroid. The segment in this case is generated by the revolution of BMP (fig. 55) about BC, and hence its volume 7r/ y-dx=ir-r^ / Jo D "Jo where B is the origin and 100. Volume of the Frustum of a Spheroid when one of the Terminating Planes passes through the Centre. (a) The prolate spheroid. The frustum in this case is generated by the revolution of BCMP about CM (fig. 23). Now volume generated by BCMP = volume generated by BCA - volume generated by PMA where 1"= CM = height of frustum = a - h . (j8) The oblate spheroid. We can show in a similar that the volume generated in this case The above formulae may be put into another form. Thus, in the case of the prolate spheroid, since the point P lies on the ellipse b-x 2 + a-y- = a-b-, we have
b-k 2 + a-b* = a-b 2 , where 6j = PM, which gives