460 



does not ; and the surface is either a paraboloid or a cy- 

 linder. 



1°. If the surface is a paraboloid, we may suppose the 

 origin of coordinates to be at its vertex, in which case both 

 H and K vanish, and we have the relations 



^^ + yi cos^ — a?!^ — ?/i^ z: ; 

 the equation of the surface being 



2/^sin2^ +g^ + 2Ga; = 0, (ig^ 



which shews that the paraboloid is elliptic, having its axis 



in the direction of x, and the plane of xy for that of its 



greater principal section. From the relations (17) we obtain 



the following, 



«/i^an2.^ +2G;ri + 6' = 0, 



y^ sin^ ^ cos^ ^ + 2Ga;2 — g^ = ; 



from which we see that the focal and dirigent curves are 

 pai'abolas, having their axes the same as that of the surface ; 

 and their vertices equidistant from the vertex of the surface, 

 but at opposite sides of it. The concavity of each curve is 

 turned in the same direction as that of the section xy. The 

 focus of the focal parabola is the focus of the section xy, and 

 its vertex is the focus of the section x% of the surface ; its 

 parameter being the difference of the parameters of these 

 two sections. The parameter of the section xy is a mean 

 proportional between the parameters of the focal and diri- 

 gent parabolas. 



2°. If the surface is a cylinder, we may make g and H 

 vanish, by taking the origin on its axis. We then have 



x^-x^, yyzzy^cos't^, 



K = yx tan^ ^ =: y^ sin^^ cos'"^ ; 



the equation of the cylinder, which is elliptic, being 



2/2 sin^ ^ + ^2 _ jj^ (21) 



Here the focal and dirigent are each a pair of right lines 



