461 



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

 foci and directrices of a section perpendicular to the axis. 

 The corresponding focal and dirigent lines lie at the same 

 side of the axis. 



II. When m = I, and ^ is not zero, B vanishes, but a 

 does not. 



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

 ordinates at its vertex, the quantities g and K vanish ; same 

 the equation of the surface becomes 



x^ tan^ ij, - z"" = 2hi/, (22) 



and we have the relations 



K = .ya — yi , a-, = X2 sec'^ ^, 

 X2^ sec'^ <p •\- 1/2 — iiO\ — !/i^ = 0. 



The paraboloid is therefore hyperbolic, its axis being that 



of y, which is also the directive axis; and as the tangent of 



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



of the focal curve, may be either of the principal planes 



passing through the axis of the surface. The relations (23) 



give 



xi^s'm'^<l>~2iit/i — n^ = 0, 



x.^ tan= sec^ ^ - 2 H^a + H^ = 0, ^ ' 



for the equations of the focal and dirigent, which are there- 

 fore 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 parabola is the focus of the section 

 xy, and its vertex is the focus of the section yz, its para- 

 meter 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 



