330 PROCEEDINGS OP THE AMERICAN ACADEMY 



= ^6l^^[si^4^-4'^]]; (37) 



where v = Sh ~ ^ -j-. 



If we revolve the curve about the diameter a, we may then put u 

 — bx, and the area of the surface generated will be found to be 



r r(a2a:2 _I_ 52)f-,x 



Xo 



= $|:[Ch«.]\ (38) 



ax 



where v = Sh — ' -^ . 



17°. Again, in 14'', we found for the parabola, when « _L |3, ' 

 TVqq' = Y ^^^ 



Whence, for the volume of the solid of revolution (generated by the 

 revolution of a sector), the axis being the tangent at the vertex, we 



a 



have, making m ^ ^ ^ j 



X X 



V= i r/.Ju.TVoo' = <D g'Jx^ = ^U f^[x^ - :r/]; (39) 



Xo Xo 



and for the volume of the paraboloid whose axis is the diameter, mak- 

 ing u =i hx, 



X 



V = e^ Jx' = cu '^1 ^x* - v] . (40) 



Xo 



18°. For the cycloid we have found, in 11°, 



Q = a{d — sin d) -\- §{l — cos 0), 



Tq' = 2a sm^^O. 



If we revolve it about the base, we shall have u = a(l — cos ^), 

 and the area of the surface generated will be 



9 e 



S= fh CuTq' = 2 0a^ fsin ^ ^ (1 — cos e) 



