On the Surfaces of the Second Order. 301 



of 2 will be the internal axis of the other cone. Let K be the angle 

 made by the two sides of the first cone which lie in the plane 

 xz, and K the angle made by the two sides of the second cone 

 which lie in the same plane ; the former angle being taken so 

 as to contain the axis of as within it, and the latter so as to con- 

 tain within it the axis of s. Then, considering r, r f as radii of 

 the section xz of the surface, we have obviously 



(3) 

 2 !' 



observing that when these formulae give a negative value for r 2 

 or r' 2 , in which case the surface expressed by the equation (2) 

 must be a hyperboloid, the direction of r or / meets, not that 

 surface, but the surface of the conjugate hyperboloid expressed 

 by the equation 



Now calling and 9' the angles made by the tangent plane of 

 the cones with the directive planes of the given surface, which 

 are also the directive planes of each cone, the angles K, K depend 

 on the sum or difference of 9 and 6'. If the latter angles be 

 taken so that their sum may be equal to the supplement of K, 

 their difference will be equal to K', and the formulae (3) will 

 become 



(5) 



*-*<-* 



by which the semiaxes of any central section are expressed in 

 terms of the non-directive semiaxes of the surface, and of the 



