VOLUME OP A SPHERE.] 



MATHEMATICS. MENSURATION. 



655 



into any number of equal parts, of which let m m 1 be 



Fig. 49. 



one, through m, draw 

 mk parallel to B O, 

 meeting A B in p, and 

 Q C in k, and through 

 TO, draw a line m l k, 

 parallel to mk, and 

 from p and k draw 

 pp, and Wij perpen- 

 dicular to m, k,. Join 

 Op. Now Op 2 = nip 1 

 -f- mO 2 since Omp is a 

 right angle. But since 

 triangle Om/i is simi- 

 lar to OACandAC = 

 AO .'.Om=mA also fan 



B O = O .'. mt 2 = mp* + mh* .'. * X 

 \f, -- 



"X 



x nA* or (Mensuration of Areas, Art. 8) the circle, 

 the radius of which is mk, is equal to the sum of those 

 whose radii are mp, and mh ; also since kk- = ppi = Wi, 

 we have 



w mk 1 X A*, = v. mp* 



+ r mA* x 



or the cylinder, the radius of whose base is mk and 

 height JUt,, is equal to the cylinder, the radius of whose 

 base is mp, and height pp,, together with that the radius 

 of whose base is mk, and height hh, ; but these cylinders 

 are those which will be described by mk, mpi and mk,, 

 when the whole figure revolves round A O. And the 

 same is true of the cylinders corresponding to any other 

 one of the equal parts into which A N is divided, and 

 therefore is true of all of them. Now all the cylinders 

 corresponding to km, make up the cylinder described by 

 A Q, and ail those described by m,;, make up the series 

 of cylinders inside the sphere, and those described by 

 m, k make up those described about the cone. Hence, 



Cylinder A Q = sum of cylinders inside portion of 

 sphere -f- sum of cylinders outside portion of cone. 



This being true, however small m m, may be, is true 

 in the limit, but the portion of the sphere is the limit 

 of the inscribed cylinders, and the frustum of cone is 

 the limit of the cylinders outside the cone (compare 

 Note, p. 646). Hence, 



Cylinder described by AQ = portion of sphere described 

 by A P N -f- portion of cone described by A C M N. 



Now let r = radius of sphere A = height of A N, and 

 V, = Tolume of portion of sphere NM = ON = r k. 



Now the column of cylinder described by A Q = ir^k. 

 And volume of frustum of cone described by A C M N 



.*. wr'h = V, + 



-/, 



.'. SVi-S* r*A rk [ i* + (r k) r + (r K)* j 



V '=1'- (*-*) 



TT'-nce if V= volume of the whole sphere, in which 

 case li = 2r. 



v- 4 ""! 3 



" 3 

 Now the volume of cylinder circumscribing sphere 



which are the faces of the solids, having the centre of 

 the sphere for their common vertex. Hence, if the 

 volume of the solids is V 



1 . 1 . 1 



/. V= - . (circumscribing cylinder). 

 3 



COR. 1. The above proposition may be demonstrated 

 in the following manner. Suppose a solid having any 

 n'imber of plane faces to be described in the sphere, and 

 let A. A 2 A s &c., be the areas of these faces, andp, p 2 p 3 

 &c., be the perpendicular distances of these faces from 

 the centre of the sphere ; now this inscribed solid may 

 be conceived to be made up of pyramids, the bases of 



Now this is true, however great the number of faces 

 maybe, and .'. is true in the limit, but in the limit, p, p 2 

 p. . . . become equal to one another, and to r the radiui 

 of sphere. Hence, 



Limit of A, p, + A 2 p a + A 3 p 3 +. . .=r x (limit 

 of A, + A 2 x A 3 ...) 



Now the limit of A , -f A 2 -f A 3 + . . . = surface of 

 sphere = 4;r r 2 (Mensuration of Areas, Art. 19). Also 

 the limit of V, = V the volume of sphere. 



COR. 2. To find the volume of the portion of the 



sphere corresponding to B P N O. Let the volume be 



called V, and let h = O N which is = M i\. Then 



V= cylinder B N cone O N M. 



Now volume of cylinder B N= 



Volume of cone M N = 5- 



/. V- 



,(, 



15. To find the Volume of a Spheroid. 



DKT. A spheroid is a figure formed by the revolution 



of an ellipse about one 

 of its axes ; if about 

 the major axis it is 

 called a prolate sphe- 

 roid, if about its minor 

 axis it is called an ob- 

 late spheroid. 



Let A Ba be a semi- 

 ellipse, O A its semi- 

 major axis, O B its 

 semi-minor axis. With centre O and radius O A describe 

 a semicircle A Co, draw Q P N through any point in 

 the ellipse parallel to O C, draw Mnm parallel to N P Q 

 and complete the rectangles M P, M Q (Mensuration of 

 Areas, Art. 12). 



Now if the figure revolve round Act the semi-ellipse 

 will describe a prolate spheroid and the circle a sphere, 

 also M P and M Q will describe cylinders with altitudes 

 M N, and having the radii of the bases N P and N Q. re- 

 spectively, and .'. having volumes TT MN X PN 2 and ir. 

 M N X N Q 2 . Now by a property of the ellipse, 



QN :PN ::o :5 

 .'. QN Z :PN 2 ::a 2 : 5* 



But Cylind. M Q : cylind. M P : : M Q 2 : M P 2 

 .'. cylind. M Q : cylind. M P : : a 2 : 6 2 



and the same is true of any other cylinders described in 

 like manner within the sphere and spheroid, and .'. is 

 true of their sum 



.'. all cylinders within sphere : all within spheroid : : a 2 : b 2 

 and this being true, however many cylinders there may 

 be, i.e. , however small we suppose M N to be, is true in 

 the limit, but the sphere is the limit of its inscribed 

 cylinders, and the spheroid the limit of its inscribed 

 cylinder, 



.'. Sphere : spheroid : : a 2 : 6 2 : : - - : 



o *> 



But volume of sphere = 

 .'. Volume of spheroid 



COR. In like manner the volume of an oblate spheroid 



3 



