On the Surfaces of the Second Order. 269 



lution of an ellipse round its minor axis, and the focal and diri- 

 gent curves are circles. 



II. When m is greater than unity ; A and B are negative ; 

 and if K be finite, it is also negative ; whence P and Q are po- 

 sitive, and K is negative. Also, supposing not. to vanish, Q 

 is greater than P. The surface is therefore a hyperboloid of 

 one sheet, with its real axes in the plane of xy ; and the direc- 

 tive axis is the primary. The focal and diligent curves are 

 ellipses. But when = 0, the surface is that produced by the 

 revolution of a hyperbola round its imaginary axis, and the 

 focal and dirigent* are circles. 



If K = 0, which implies, since A and B have the same sign, 

 that #1, y t , #2, y* are each zero, the surface is a cone having the 

 axis of z for its internal axis ; and the focal and dirigent are 

 each reduced to a point. The focus and directrix are conse- 

 quently unique ; the focus can only be the vertex of the cone ; 

 the directrix can only be the internal axis ; and the directrix 

 therefore passes through the focus. The directive axis, which 

 coincides with the axis of y, is one of the external axes ; that 

 one, namely, which is parallel to the greater axes of the elliptic 

 sections made in the cone by planes perpendicular to its internal 

 axis. This is on the supposition that is finite ; for, when 

 = 0, the cone becomes one of revolution round the axis of z. 



III. When m is greater than cos 0, but less than unity, we 

 have A positive and B negative, and the species of the surface 

 depends on K. It is inconsistent with these conditions to sup- 

 pose = 0, and therefore the surface cannot, in this case, be one 

 of revolution. The value of K may be supposed to be given 

 by the formula 



K= Ll (^ - *,) + i? (y, _ y l)% 



which contains only the relative co-ordinates of the focus and 

 the foot of the directrix, and is a consequence of the equations 

 (6) and (7). 



* When the term dirigent stands alone, it is understood to mean a dirigent line. 



