240 PROCEEDINGS OF THE AMERICAN ACADEMY 



— 2 dSia^xa^ = -\- x 2d=z c, 



+ c 

 X = — • 

 2d 



Now substitute 



P = ^ + 2^ "i' 



and we find 



(J 

 c^SV,? ~f~ ^^sS'WsP — 2 dSa^Q -f- 2 c? ^-, = c, 



CgS'^ttg? 4" ?3S^«3(? — 2 f?SrtjP = 0. 



The quadric represented by this equation is an elliptic or hyperbolic 

 paraboloid, according as the c's are alike or unlike in sign. Because if 

 the surface be cut by a plane 



SttgO = c or SwgO = c 

 perpendicular to either a.^ or «., the section is 



c^S^a.,Q = 2 dSa^Q — c, 

 or 



CgS^ttgO = 2 dSu^Q — c, 



either of which is easily seen to be a parabola. But if cut by a plane 

 perpendicular to a^ 



S«j« = c, 

 the section is 



c.^^^a.^n -j- c^S-a^Q = — c, 



which is an ellipse or hyperbola, according as c^ and c^ are alike or 

 unlike in sign. The sign of c shows the side of the origin in which 

 the cutting plane lies, and determines whether the elliptic section is 

 real or imaginary, or whether the hyperbolic section has its transverse 

 axis parallel to Wj or to a^. 



We have considered the case in which 



c, = 0, 

 and 



y = — du^ — ga^ — ha^ 



Let us now go a step farther, and suppose d, g, or Tc to vanish, and let 

 us first take it as the one corresponding to the root that has disap- 

 peared. In this case 



and therefore 



