455 



Here we cannot suppose k to vanish, as the surface would 

 then be reduced to a point. 



When = 0, that is, when the directive planes coincide 

 with each other, and therefore wifeh a plane perpendicular to 

 the directrix, so that SD is the shortest distance of the point 

 S from the directrix, the surface is a spheroid produced by 

 the revolution of an ellipse round its minor axis, and the 

 focal and dirigent 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 R is negative. Also, supposing <j> 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 

 directive 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 Xi, yi, x^, ^2 are each zero, the surface is a cone 

 having the axis of « for its internal axis; and the focal and 

 dirigent are each reduced to a point. The focus and di- 

 rectrix are consequently 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 be- 

 comes 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 



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

 line. 



VOL. II. 2 R 



