Conic Sections. 263 
rive, from a familiar property of the hyperbola, a? : 5? ::c0s.u/? 3 
If sin.w? > 
; : b? ‘ : 14 
Sin.w'*.".sin.a!? = 7 ry a?4b3 (G) is negative, and 
2 
if sin <p (G) is positive. Wherefore we conclude that 
the section of an hyperboloid parallel to an asymptote is a para- 
bola; that, if the section makes a greater angle with the axis of the 
hyperboid than the asymptote does, it is an ellipse or circle ; and 
that, if the section makes a less angle, it is an hyperbola. To de- 
termine the sections of the paraboloid, we put c=0, and therefore 
G= —a?*sin.w?, which is zero when w =O, and negative in all other 
cases. Hence it is inferred that the section of a paraboloid parallel 
to its axis is a parabola, and that in all other positions itis a circle or 
an ellipse. In the sphere and spheroids c= —1 and G=—6?— 
(a? —6*)sin.w? ; which, being always negative indicates that all the 
sections of the sphere and spheroids are circles or ellipses. The 
sections of a cone are determined with equal facility. ‘The sections 
of a cylinder may be derived directly from (3) or (4) by putting 
p=0, c=1, and considering the axis of revolution C’K’ (Fig. 2,) 
to be parallel to ARH, which in this case is a right line. All the 
parabolic sections of the solids under consideration being charac- 
terized by the equation y? = =%, have for their parameter. 
6? (p+2cd)cos.w —2ab/ (pdt cd? )sin.w 
All other sections, ex- 
cepting the right line are characterized by (10) without any change 
of general form. Putting f= and = =" as (the +-or— be- 
ing used according as (G) is positive or negative) (10) becomes 
S Agee =y?, or aa, 2)=y?, the equation of a cir- 
cle, ellipse, or hyperbola whose principal axis is A=-+ 
b? (p+-2cd)cos.w — 2ab(pd+-cd?)sin.w ; ; : 
cb? — (a? +-cb)sin.w? » and its conjugate axis B= 
b9(p-+2ed)cos. w — 2abs/ (pd+-ed?)sin. w 
fe ma +cb*)sin.w?)} 
Be eee 3 a? 2: [cb? — (a? +-cb?)sin.w*] ; which is an 
Wherefore also A? ; 
