260 Conic Sections. 
RVT being each perpendicular to the plane ARHM, their common 
section VT is also perpendicular to RV and HK ; and consequently 
HK? —VK*=VT?=y’. But VK=RQ+ VP=7-+sin.(a—9)x- 
Substituting in the last equation this value of. VK, and for HK the 
value (w’) in (2), we shall have 
e+ fcos.(w—o)x (ict a) * P 
( oa Qn =\\ on c 
= 
h+dcos.(w — 9) x-+mcos.(w —9) *x*) , (3-+sin.(—e)x). (3); 
n 
1 
which by an obvious reduction becomes y* = 77> & — feos.(w—o)x 
‘ 1\2 
[e? —4nh+2(ef —4nd)cos.(u—o)y +(f2—4mn)cos.(u — 9)? x? | J = 
[x-+sin.(w —o)x]?, (4); which are the equations, referred to rec- 
tangular coordinates of the section of a plane with the surface of a 
solid formed by the revolution of any conic section about an axis 
situated in its plane. 
The equations (2), (3), (4), are liable to a failing case which, 
though of rare occurrence, it may be well to notice and make pro- 
a? 
a? + ch?’ 
casioned by regarding (1) as a quadratic equation, which it evidently 
ceases to be when n=0. In this case 
h+dcos.(« —¢)x-+mcos.( — 9) 2x? 
vision for. This happens when n=0 or sin.o? = and is 0c- 
w= a} fetta = oy —, (5); and consequently 
: ({h+5ceos.(w —0)y-+-mcos.(w —o)*x? 2 , t 
i \ e+ feos.(a —o)x —[x+sin.(«-9)x]? (4 ): 
In the application of these general equations to proposed cases it 
is sufficient merely to remark that (K), (4), (#), (v), are positive in 
the situations in which they are represented in fig. (1), and change 
their signs according to the familiar principles of trigonometry. !* 
is evident from an inspection of the figure that (+) and (d) are not 
arbitrary lines, being dependent upon the equation of ARH and the 
given magnitudes (k), (8), (p). To determine (7), it is necessary 
only to put (7) in the place of (w’) and zero in that of (t’) in (2) 
which expresses the relation between the rectangular coordinates of 
ARH relative to the axis CK and origin of abscisse Q. This sub- 
Ba tee e e2  h\3. 
stitution gives *= 5 +(75—") ; which fails as above when 
