On the Surfaces of the Second Order. 275 



curve, may be either of the principal planes passing through the 

 axis of the surface. The relations (23) give 



x? sin 2 - 2Hy! - H* = 0, 



(24) 

 tan 2 sec 2 - 2Hy 2 + H* = 0, 



for the equations of the focal and dirigent, which are therefore 

 parabolas, having their axes the same as those of the surface, 

 and their concavities turned in the same direction as that of the 

 section xy ; their vertices being equidistant from the vertex of the 

 surface, and at opposite sides of it. The focus of the focal para- 

 bola is the focus of the section xy, and its vertex is the focus of 

 the section yz, its parameter being the sum of the parameters of 

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

 proportional between the parameters of the focal and dirigent 

 parabolas. 



2. If the surface is a cylinder, and the origin on its axis, 

 G and H vanish, and we have 



o; = ^sec 2 =^2, 



(25) 

 sec 2 ^ ; 



the equation of the cylinder, which is hyperbolic, being 



x* tan 2 - s 2 = - K. (26) 



The focal and dirigent are each a pair of right lines parallel 

 to the axis of the cylinder; the corresponding lines passing 

 through a focus and the adjacent directrix of any section per- 

 pendicular to the axis. The directive planes are parallel to the 

 asymptotic planes of the cylinder. 



In this case, if K = 0, the surface is reduced to two directive 

 planes, and the focal and dirigent to the intersection of these 

 planes. 



III. When m = 1, and $ = 0, both A and B vanish, and the 

 surface is the parabolic cylinder. If, as is allowable, we sup- 



T2 



