Plane Sections of a Cone of Revolution. 187 
Whence 
cos (2 — DCK ) =e cosa 
DCK = — 6 
ECK = 6+4 
Therefore any plane making with the axis of the cone the 
same angle as does CH, such for example as X, X, (drawn 
parallel to CJ/f ) will cut from the cone a hyperbola of the 
given eccentricity. 
' It is to be noticed that if CZ, the radius of the second 
arc, is taken greater than CD = , the limit of the eccen- 
1 
C08 & 
tricity, the perpendicular to the axis of the cone drawn as 
above will not intersect the first arc; the construction is 
therefore impossible, and no section of the cone with such 
eccentricity exists. 
Also, the section of maximum eccentricity is shown by the 
construction to be the section cut from the cone by a plane 
parallel to the axis. 
or the parabolic section, e = 1,@ = 0, and C& coincides 
with CD, an element of the cone. 
Therefore, any plane making with the axis the same angle 
as do the elements of the cone will cut from the cone a 
parabola. 
(6) The formula generally used to express the /atus 
rectum of a conic is 
bo 
26 
So sees to (2) 
in which 
2 p = the latus rectum. 
2 a = the transverse axis. 
2 6 = the conjugate axis. 
The equation of a conic referred to its vertex and in terms 
of its semi-axes is 
3? 
y? = qi (2 2% = 2?) (3) 
