CONIC SECTIONS. 



parallels at right angles. Because T S is 

 perpendicular to P Q, therefore M N is 

 perpendicular to D V (parallel to P Q) ; 

 but M N is also perpendicular to D A : 

 therefore it is perpendicular to the plane 

 D A V (4. 11. E.) : therefore A V B is a 

 section of the cone through the axis at 

 right angles to the base (18. 11. E.) 

 Again, because the section is a circle, 

 therefore PRxRQ=SRxRT: con- 

 sequently V D> = A DxD B (Pr. 6.) 

 Therefore V D is a tangent of the circle 

 described about the triangle A V B, and 

 the angle DVB is equal to the angle 

 A V B (32. 3. E.) Therefore the circular 

 section is a subcontrary one. 



Cor. No other than a parallel and asub- 

 contrary section of a cone is a circle 



Fig. 12, 13, 14. If a cone be cut by a 

 plane P Q, which neither passes through 

 the vertex, nor is parallel to the base, 

 then a plane, as V M N, being drawn 

 through the vertex parallel to the cutting 

 plane, it will necessarily meet the plane 

 of the base of the cone. The line of 

 common section of the parallel plane, 

 and the base of the cone M N, may have 

 one or other of these three different posi- 

 tions, viz. 



1. It may be without the base of the 

 cone. 



2. It may touch the periphery of the 

 base. 



3. It may cut the periphery of the 

 base. 



These three different cases offer three 

 sections for our consideration, that are 

 very different from one another, and pos- 

 sess many properties peculiar to each, 

 while they have many common to all the 

 three. 



Def.5. Fig. 12. If the line of common 

 section M N be without the base of the 

 cone, then the plane V M N drawn 

 through the vertex will be entirely be- 

 tween the two conic surfaces, not meeting 

 either of them. In this case the cutting 

 plane P Q will meet every line drawn in 

 one of the conic surfaces, and the curve 

 line of common section will surround that 

 conic surface, and will completely inclose 

 a space. In this position of the cut- 

 ting plane, the line of common section, 

 unless when it is a circle, is called an el- 

 lipse. 



Def. 6. Fig. 13. If the line of common 

 section MN, touch the periphery of the 

 base of the cone, then the plane drawn 



through the vertex will touch the conic 

 surfaces (.Pr. 3,) and the opposite surfa- 

 ces will be on opposite sides of it. In 

 this case the cutting plane will meet eve- 

 ry line drawn from the vertex in one of 

 the conic surfaces, excepting only the 

 line VB, in which the touching plane 

 meets the conic surface ; and as the cut- 

 ting plane is indefinitely extended along 

 the touching plane without meeting it, 

 it is obvious that the curve line, formed 

 by the common ' section of the cutting 

 plane and the conic surface, does not re- 

 turn into itself so as to inclose a space, 

 but is open on the side opposite to the 

 vertex of the cone. In this position of 

 the cutting plane, the conic section is 

 called a parabola. 



Cor. 1. Every right line drawn in the 

 plane of a parabola, which meets the 

 curve in one point, but neither touches 

 the curve, v see Def. 8,; nor is parallel to 

 the line VB in the conic surface, will 

 meet the parabola again in another point. 

 This is manifest from Prop. IV. 



Cor. 2. All right lines drawn in the 

 plane of a parabola, which meet the curve 

 in one point only, but are not tangents, 

 are parallel to one another. For they are 

 all parallel to the line VB in the conic 

 surface. (Cos. 1.) 



Def. 7. Fig.l4i. If the line of common 

 section MN cut the periphery of the 

 base, then the plane drawn through the 

 vertex will divide each of the opposite 

 conic surfaces into two parts lying on op- 

 posite sides of it. In this case the cut- 

 ting plane, being indefinitely extended, 

 will meet every line drawn from the ver- 

 tex in those parts of the two conic surfa- 

 ces that lie on the same side of the plane 

 through the vertex as the cutting plane 

 itself; and thus two curves will be form- 

 ed by the common intersection of the 

 cutting plane, and the two opposite co- 

 nic surfaces. It is obvious that these 

 curve lines may be indefinitely extended, 

 and that they do not return into them- 

 selves so as to inclose a space. In this 

 position of the cutting plane, the conic 

 section formed by its intersection with 

 one of the conic surfaces is called a hy- 

 perbola ; and the two conic sections form- 

 ed by its intersection with the two oppo- 

 site conic surfaces are called opposite 

 hyperbolas, or opposite sections. 



Cor. 1. Let m V n be the common sec- 

 tion of the cone, and a plane drawn 

 through the vertex parallel to the plane 

 of the two hyperbolas : then every right 

 line drawn through a point of one of the 

 hyperbolas, so us to be parallel to either 



