322 A r ote on Surfaces of the Second Order. 



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

 whose equation is 



ZT2 2 V'i 



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 S is 

 equal to OT, while it 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 



^T 17 ' ~ J ~ ' w 



iff iw K 



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 



r 



p 



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

 tively, we have 



1_ 1 .1 1_ 1 _1 



p ~ z* + r' p' ~ 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 S with a force varying inversely as the fourth power of 

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



