Conic Sections. 261 
; voile 
nm=0; in which case r= — 22 as is evident by putting x=0 in (5). 
If the axis of revolution is parallel to the principal axis of the re- 
volving curve, 9=0, sin.p=0,cos.o=1, k= d,h=a?d*-b?(pd+ced?), 
6=—62(p+2cd), e=2a?6, f=0,m= <b, n=a?, anda=—/(pd+ 
cd?) —6é. Substituting these values, equation (3) is reduced to y? 
b 1\2 /b 
(1° (pd-ted? + (p+2cd)cos.wx + c.cos.w? x? t - (- V (pd+ 
ed?) —8-+-sin.v | ‘ (6). The most simple as well as useful class of 
cases embraced in (6) is that in which the axis of revolution is sup- 
posed to coincide with the principal axis of ARH. According to 
b? 
this hypothesis 6=0, and becomes aa ae (pated +(p+ 
2cd)cos.wyx + c.cos.w? x ied ~~(pd-+ed*) 2 (pd +cd?)sin.wy — 
2—(a? eee 
sin.w*y%?, or yt 
b? (p+2cd oe w— QabV/ (pd+cd? )sin.w 
(! ipgho4) “(a by a ley x3) (7) ; which is evi- 
dently a conic section, and characterizes the section of a plane with 
a cone, sphere, the spheroids, hyperboloid, or paraboloid, according 
as ARH is a right line, circle, ellipse, hyperbola, or parabola. If, 
for brevity, we put cb? — (a? +cb?)sin.w? =G (8), and b2(p+2cd) 
cos.w —2ab,/(pd+cd*)=gq (9), the equation of the section becomes 
y? uSi(m +x | (10). Conceiving A (Fig. 2,) to be that prin- 
cipal vertex of the solid which is nearest to R, it is evident that 
(p+2cd) is always positive ; for (c) is negative only in the sphere 
and spheroid, in which cases c= —1, p=a, anda>2d. When the 
plane RV is a tangent to the solid at R, SL: RL: :cos. w : sin. w. 
But, according to well known properties of the conic sections, the 
2(pd+ced? 1”) b fs 
subtangent ares q_ > and the ordinate RL=_ Vv (pdt+ed ). 
Substituting these values in the proportion above and reducing, we 
have 6°(p+2cd)cos. w—2abs/(pd+cd? )sin. w=0. As RV de- 
scends cos. « and sin. w continue positive, the former increasing and 
the latter decreasing till they become respectively (1) and (0) when 
RV coincides with RF. When RV is siwated between RF and 
