270 On the Surfaces of the Second Order. 



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, in the directive, but the focal and diri- 

 gent are now hyperbolas. 



2. If K is a negative quantity, the surface is a hyperboloid 

 of two sheets, having its primary axis coincident with that of oc. 

 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 x for 

 its internal axis ; the directive axis being, as before, that exter- 

 nal axis to which the greater axes of the elliptic sections, made 

 by planes perpendicular to the internal axis, are parallel. The 

 axis of & 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, from 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 and direc- 

 tive axes ; and since by the equations (6) and (8) the value of K 

 may be written 



K = x l (x z - Xi] + y l (y z - yj, 



we see that, as K now vanishes, the right line joining corre- 

 sponding points F and A upon these lines is perpendicular to 

 the focal line. Of the two sides of the cone which are in the 

 plane xy, one lies between each focal and its dirigent ; and 'it 

 may be inferred from the equations, that the tangents of the 

 angles which the internal axis makes with a focal line, with one 

 of these sides of the cone, and with a dirigent line, are in con- 

 tinued proportion, the proportion being that of the cosine of 

 to unity. And hence it follows, that these two sides of the cone, 



