264 Conic Sections. 
invariable ratio when (w) is constant, and shews the similarity of all 
the parallel circular, elliptic and hyperbolic sections of these solids. 
If a=b=p and c= —1, (6) is reduced to y =( —é+[ad —d?+ 
(a — 2d)cos.uy —c08.u 2? | *} ae [/ (ad —d?) —é+sin.wy]? (11), 
which characterizes the sections of a solid formed by the revolution 
of acircle about any axis parallel to a chord, and embraces those 
curves imagined by Perseus Citicus denominated spiriques. If in 
this case 6=0, the solid is a sphere ; if CK is above AL (8) is pos- 
itive, and if below, negative. If ¢=—S the axis of revolution is 
a tangent to the generating circle. 
a 
If 6> 9 (11) is the equation of the section of a circular ring. 
Ifin (6) a=p and c=—1 (while (a) and (4) are unequal, y?= 
b b 
: Lamia (ad d? +-(a —2d)cos.wy —cos.w? x? )3) ae Es J (ad — 
d*)—b-+sin.wy) (12) which the equation of the sections of an 
elliptical ring which is elongated or flattened towards the axis of 
revolution according as (a) is greater or less than (b). The equa- 
; b 
tion (6) is reduced to y2= ( i (aa+ d? +- (a+2d)cos.uix + 
1 b 
cos. wy?) ) "(EV (add?) —4sin.y) (13), and y7= ( ~ b+ 
v (pd+-pcos. x) oe a) —+sin.ury ° (14), according as 
ARH is an hyperbola or parabola. 
If the axis of revolution C’” Q” is perpendicular to the principal 
AL, 9=90°, sino=1, coso=0, AC’=k, RQ”=d—k="; 
b 
Q” C”=t=RL =—v (pd+cd?), h=b?(pd+cd?)-b? (pk+ck*)s 
6=2ab/ (pd+cd?), e=b? (p+2ck), ‘f=0; m=a?, and n= —¢b*. 
Substituting these values in (3) we derive the equation 
