430 



From the form of this equation it is evident, that if the 

 surface be intersected by the plane whose equation is 



SOS , Vo^ , 4tl4 1 , 7 X 



it will be touched along the curve of intersection by the cone 

 whose equation is 



T+F + F = ^- (^> 



This mode of deducing, in its simplest form, the equation of 

 a cone circumscribing a surface of the second order, is much 

 easier than the direct investigation by which the equation (c) 

 was originally obtained. 



Let a right line passing through S intersect the plane 

 expressed by the equation (6), in a point whose distance from 

 S is equal to w, while ft intersects the surface A in two points, 

 P and P', the distance of either of which from S is denoted by 

 p. Let the surface B, represented by the equation 



P TJ^ P 



be intersected by the same right line in a point whose distance 

 from S is equal to r, the distance r being, of course, a semidia- 

 meter of this surface. Then it is obvious that the equation (a) 

 may be written 



i - 1^1 _ r 



r^ ~ \v7 p/ 



so that, if p and p' represent the distances SP and SP' respec- 

 tively, we have 



111 111 , . 



p ts r p' -m r ^ 



and therefore 



1 I 2 



