430 
From the form of this equation it is evident, that if the 
surface be intersected by the plane whose equation is 
E on Cake 
= aii Sopa aig ae (2) 
z 
22, 
k 
it will be touched along the curve of intersection by the cone 
whose equation is 
£2 n? la 
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 (d), in a point whose distance from 
S is equal to a, 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 
ae aes 
be intersected by the same right line in a point whose distance 
from S is equal to 7, the distance 7 being, of course, a semidia- 
meter of this surface. Then it is obvious that the equation (@) 
may be written 
Lay y 
Pi. 6 BI. 
so that, if p and p’ represent the distances SP and SP’ respec- 
tively, we have 
; (e) 
and therefore 
(YS) 
