OF ARTS AND SCIENCES. 333 



S = ^b I Shxs/c-Ch-'x — a' 



2c L a Y a2 a J ,^ 



= ^^lSh2v — 2vT , (46) 



4c L -i vc 



where v =z Ch~^ 



P 



cChx 



a 



22*^. By the application of formula (7) the volume generated by 

 the revolution of a sector of a conic about a principal axis may be 

 obtained very simply. With a X /3, the equations of the ellipse and 

 hyperbola give 



TVqq' — TV«^ = ab. 



The substitution ti =:z b sin x gives 



X 



F= ^ ^ CuTVqq = ^ ^ ab'' fsin x 



= ^ ^ ab^ \ COS x\ , (47) 



for the volume enclosed by the prolate ellipsoid ; and the substitution 

 u = a cos X, gives 



X 



V= ^ ^ a-b I COS X = ^ ^ a'^bl sin a: , (47') 



for the volume enclosed by the oblate ellipsoid. 

 23*^. Similarly, the substitution u = aChx gives 



X 



V=^"^ Cchx =^"^ [Sh:c] * , (48) 



Xo 



for the volume swept by a central sector in generating the unparted 

 hyperboloid ; and the substitution u :=;: bShx gives 



X 



Xo 



for the volume swept by a central sector in generating the parted 

 hyperboloid. 



