456 



surface depends on k. It is inconsistent with these condi- 

 tions to suppose = 0, and therefore the surface cannot, in 

 this case, be one of revolution. The vahie of k may be 

 supposed to be given by the formula 



K := 1^ (a, - a^.y + 1^ (y, - y,)^ (1 4) 



A B 



which contains only the relative coordinates of the focus and 

 the foot of the directrix, and is a consequence of the equa- 

 tions (6) and (8). 



1°. If K is a positive quantity, the surface is a hyperboloid 

 of one sheet, with its secondary axis in the direction of x ; 

 the primary axis, as before, is the directive, but the focal and 

 dirigent are now hyperbolas. 



2°. If K is a negative quantity, the surface is a hyperbo- 

 loid of two sheets, having its primary axis coincident with 

 that of or. The secondary axis is the directive; the focal 

 and dirigent are hyperbolas. 



3°. If K = 0, the surface is a cone, having the axis of ar 

 for its internal axis; the directive axis being, as before, that 

 external axis to which the greater axes of the elliptic sec- 

 tions, made by planes perpendicular to the internal axis, are 

 parallel. The axis of z is the other external axis, which 

 may be called the mean axis of the cone, because it coincides 

 "with the mean axis of any hyperboloid to which the cone is 

 asymptotic. As a and b have different signs, it is evident, 

 fi'om the equations (6) and (7), that the focal and dirigent 

 are each a pair of right lines passing through the vertex, 

 each pair making equal angles with the internal axis. Two 

 planes, each of which is drawn through the mean axis and a 

 dirigent line, are the dirigent planes of the cone. 



The corresponding focal and dirigent lines are those 

 which lie within the same right angle made by the internal 

 an4 directive axes; and since by the equations (6) and (8) 

 the value of k may be written 



