322 Note on Surfaces of the Second Order. 



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

 whose equation is 



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 8 intersect the plane ex- 

 pressed by the equation (#), in a point whose distance from 8 is 

 equal to or, while it intersects the surface A in two points, P and 

 P', the distance of either of which from 8 is denoted by p. Let 

 the surface It, represented by the equation 



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

 from 8 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 



T 1\ 2 



so that, if p and p represent the distances SP and 8P / respec- 

 tively, we have 



1 1 1 j. _ 1 _1 n 



p "w * P p' ~ v r ; 

 and therefore 



This result is useful in questions relating to attraction. For 

 if A be an ellipsoid, every point of which attracts an external 

 point 8 with a force varying inversely as the fourth power of 

 the distance, and if the point 8 be the vertex of a pyramid, one 



