264 Conic Sections. 



invariable ratio when (w) is constant, and shews the similarity of all 



the parallel circular, elliptic and hyperbolic sections of these sohds. 



If a=b=p and c= — 1, (6) is reduced to y^={—6±[ad — d^-{' 



which characterizes the sections of a solid formed by the revolution 



of a circle about any axis parallel to a chord, and embraces those 



curves imagined by Perseus Citicus denominated spiriques. If in 



this case ^=0, the solid is a sphere ; if CK is above AL (&) is pos- 



a 

 itive, and if below, negative. If ^= — o the axis of revolution is 



a tangent to the generating circle. 



If ^> ^j (II) is the equation of the section of a circular ring. 

 If in (6) a=p and c= — 1 (while [a) and (b) are unequal, y^ = 

 {— &±-iad — d^ -\-{a —2d)cos.ux~^'^^''^^X^ 1 2 j _ (-^ ^^^ _ 



d^) — ^ + sin.wx| (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 (6). The equa- 

 tion (6) is reduced to y^ = i—&di:i-i ad -{- d^ + (a+2c?)cos.wx + 



cos. w^x^) M '-(-^/(fl^+c^2)_^+sin.wx] (13), and y^ = (^-6±: 



'^{pd+pcosMx)) —i\/{pd) — ^ + sm.ux] (14), according as 



ARH is an hyperbola or parabola. 



If the axis of revolution C" Q,''' is perpendicular to the principal 



AL, 9=90°, sin.(p = l, cos.9 = 0, kC'^^lc, RGl'''=d-k==^, 



b 

 Q'" C'''=&=R'L=-\/{pd-\-cd^),h=b^{pd-i'Cd^)-b^{pk+cJc^), 



S=2abx/{pd-\-cd^), e=b^{p-^2c'k), f=0, m=a^ , and n=—cb^. 

 Substituting these values in (3) we derive the equation 



