OF ARTS AND SCIENCES. 331 



4" 



= 8 0a2 Ain^ ^6 = B^a^fi cos' ^ ^ — cos i ^1 . (41) 



04 

 The complete area h S = — Qa^. 



The area of the surface formed by revolving the cycloid about |3 

 would be found by puttmg u = a(d — sin d) and using formula (5) 

 as before. 



19°. The volume generated by revolving a sector of the cycloid 

 about its base is 



e 



V=~ Cu.TVqq' = (D ^ f(l — cos d)(d sin ^ + 2 cos d — 2) 







= (Ij ^ A^ sin ^ + 4 COS ^ — i 5 sin 2 ^ — 2 cos^ d — 2) 



= (I)^\osm0—^sm2d — 6(cosd — lcos2d-\-3)y- (42) 

 The complete volume is 5 Q'^a^. 



20°. In 7°, we have found for the ellipse 



T^' = ^(a^ — b-) sin'^ x -{- b^ = y'a^ — (a- — 6'^) cos- x. 



The prolate ellipsoid is generated by revolving the ellipse about its 

 major axis. In this case, the equation being q =. a cos x -\- ^ sin x, 

 we shall have 



« = J sin X, 

 and for the surface 



X X 



S =(I) j u.Tq' = m j sin xs/d' — (a^ ^ b') cos^ x 



Xo Xo 



COS X 



=1 ^b I sjd' — c'^ cos^ X, 



COS Xo 



where c^ = d^ — b^. Whence 



_ ^ , rcos X a^ . , r cos xi " 



^=$^[_^«2_,.,,3.,_^_sin-i^] 



