332 PROCEEDINGS OF THE AMERICAN ACADEMY 



^ d'b re cos x c- cos' x , . , 

 = $— V/1 h sm-1 



c cos X 



'—, » 



T i • 1 C COS X ., 



Let y = sm~i ; then 



^=$^'[sin2. + 2f]'. (43) 



For the oblate ellipsoid of revolution, the substitution 

 u = a cos X 

 gives, by the same method of proceeding as before, 



V sinx 



S =^ ^a I cos x\Jc'- sin'^ a; -j- Z»- = $o I y/c'- sin- x -|- 



, rsin -T /'2 csinx"]'" 



= ^« L~2~V^^' ^^"' ^ + ^' + 2^ ^ ~T~J. 



, ai^ re sin x /c'^ sin- x , , , _, e sin a-i^ 



= $ \/- ■ 4- 1 + Sh — 1 • 



2c L 6 V b-2 ~ ^ b J: 



62 



^ ^, , c sin X 



Let V = bh — -^ — ; — ; then 



S=^—{sh2v-\-2vT. (44; 



21°. From the equation of the hyperbola we found (8^), 



To' = v/c^Sh^x -\-b' = v/c-Ch-x — a% 



where c^ = a^ -\- b\ Hence, for the unparted hyperboloid of revolu- 

 tion, generated by revolving the hyperbola about the axis ^, we put 



u = aChx, 

 and find 



X X 



S = <P I n.To' = ^a I ChxsJc^Slrx + b^ 



xq Jo 



= 9 — V/ — ^^ h 1 + Sh — 1 —. — 



= ^"-^[Sh2v-\-2v']\ (45) 



where u = Sh~"^ 



b ' 



Again, the substitution ic = bShx gives, for the area of the parted 

 hyperboloid of revolution. 



