OF ARTS AND SCIENCES. 337 



where v = Sh ~ ^ -f- ; 



be ■ 



v. = IV'. with a different value for s ; and finally VI. gives q\ = 

 Gx -\- a, (/^ = i o'x'^, whence 



VpV, = ^/^ + «^'^. 



and 



y X y 



where v = Sh"~i ^. The equations for the surface generated by 



revolving the parabola about its tangent at the vertex, will be found 

 from VI. and VI'. by putting b = c := s, and its finite area is 



«=[y-y.]6^[si.4»-4.];, 



where v = Sh"^ — , 



25°. Various forms of equations may be used for the quadrics. 

 Thus, the parted hyperboloid represented by III. may also be repre- 

 sented by the equations 



Q = ^S\\x -\- crCha?, 



a = aChy -\- 7 Shy. 



Again, the unparted hyperboloid represented by IV. can also be rep- 

 sen ted by 



Q = § cos a: -j- cr sin x, 



G = «Sh?/ -1- ^Chy, 

 or by 



Q = zh /3Chx -\- aShx, 



a = aChy -\- j'Shy. 



But, with regard to the last two sets of equations, it is to be observed 

 that they must be taken together, in oi-der to give a complete repre- 

 sentation of the surface, for real values of the variables x and y. For, 

 if we use only real values of the variables, the first set of equations 

 gives no points of tlie surface exterior to a pair of planes parallel to 

 VOL. XIII. (n. s. v.) 22 



