274 On the Surfaces of the Second Order. 



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 xz of the surface ; its parameter being the 

 difference of the parameters of these two sections. The para- 

 meter of the section xy is a mean proportional between the para- 

 meters of the focal and dirigent parabolas. 



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

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



X t = X l , /! = y a COS 2 0, 



(20) 

 K ' = y* tan 2 ^ = y? sm 2 $ cos 2 ^; 



the equation of the cylinder, which is elliptic, being 



3/ 2 sin 2 + s 2 = K. (21) 



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

 parallel to the axis of the cylinder, and passing through the foci 

 and directrices of a section perpendicular to the axis. The cor- 

 responding focal and dirigent lines lie at the same sides of the 

 axis. 



II. When m = 1, and $ is not zero, B vanishes, but A does 

 not. 



1. If the surface is a paraboloid, and the origin of co-ordi- 

 nates at its vertex, the quantities Gf and K vanish ; and the 

 equation of the surface becomes 



(22) 

 and we have the relations 



-H" = # - yif %\ = #2 sec 2 </>, 



(23) 

 # a 2 sec 2 ^ + y* - ati* - y\ = 0. 



The paraboloid is therefore hyperbolic, its axis being that of y, 

 which is also the directive axis ; and as the tangent of may 

 have any finite value, the plane of xy, which is that of the focal 



