

ANALl '- iliY 



[C* IV. 



f..r its loens ;i surface as represented in Fig. 101. This 

 quadrie is tin- hyperbolic paraboloid, and may be conceived as 

 generated by a variable parabola which has its vertex upon 

 and moves always perpendicular to a fixed parabola, the axes 

 of tin* t\\" parabolas being parallel, but lying in opposite 

 diivrtions. liquation [38] may be derived at once from 

 this dHinition.* 



F.\vrv equation of the form Ac 3 By* 2 Nat = repro 

 senls an hyperbolic paraboloid. 



232. The cone: equation 



I_*:=o. 

 6* c* 



The equation 

 *- = evidently is sat- 



isfied by the coordinates of only 

 one real point, viz. the origin. 

 No further discussion of this 

 equation is necessary. But tho 

 equation 



3 + S-3-0- [] 



has a locus of importance, hav- 

 ing the following prop.-r; 



(1) The origin is a point of 

 the locus. 



(2) The trace on the xy* 

 plane is a point. The traces 



on planes parallel to the a-y-plane are similar ellipses, whose 

 semi-axes increase continuously and indelinitely as z increases 

 from to oo. 



(3) The trace on each of the other coordinate planes is a 

 pair of straight lines whieh intersect at the origin. 





See Art. 228. 



