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 dirigent 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 a?!, i/i, o? 2 , 2/2 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 *. 



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= L^ (^ - -,.,). + i^? (y z - ytf, (14) 



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. 



