336 MATHEMATICS. 



elements ; and in a right circular cone, they all form equal angles 

 with its axis. Every section of such a cone, by a plane parallel to 

 its base, or perpendicular to its axis, is a circle, as EF ; which curve 

 is therefore one of the conic sections. If the cutting plane be ob- 

 lique to the axis, but make with it a greater angle than the elements 

 do, then the section will be an ellipse, as GH ; which is a curve 

 returning to itself like a circle, but elongated in one direction. If the 

 cutting plane make with the axis the same angle that the elements 

 do, the section will be a parabola, as UK : but if it make with the 

 axis a smaller angle than this, the section will be a hyperbola, as 

 LMN, OPQ. The ends, or branches of a parabola, or hyperbola, 

 never meet, but go on diverging to an infinite distance. 



If we suppose the elements of the cone CVD to be prolonged 

 beyond the vertex, they will form another cone, AVB, equal and 

 opposite to the first ; both having a common vertex. These two, in 

 connection, are technically called a cone of two nappes. If we con- 

 sider them both as extended to an infinite distance from the vertex, 

 the plane which cuts out a hyperbola from one of them, will cut out 

 an equal and opposite hyperbola from the other ; and these two are 

 called conjugate hyperbolas. A cylinder, may be regarded as a 

 cone, whose vertex is at an infinite distance from its base : and its 

 sections, by planes, whether circular or elliptical, belong therefore to 

 the conic sections. All the conic sections may be comprehended in 

 one general equation ; by varying the terms of which it is made 

 applicable to every particular case. 



In the circle, Fig. 12, if we take two diameters for the axes of 

 coordinates, and consider any point C, on the circumference, its ordi- 

 nate, CD, will be the same line as the sine, and its abscissa, OD, 

 as the cosine, of the arc between this point and the diameter which 

 is made the axis of abscissas. Calling the radius, R ; the abscissa, x ; 

 and the ordinate, y ; the equation of the circle will be y* -f x z = 7? 3 ; 

 in which x, and y, vary for the different points of the circumference ; 

 y diminishing as x increases, but R remaining constant for all points 

 of the same circle. In the ellipse, Fig. 17, the longest of all the 

 diameters is at right angles to the shortest ; the former, being called 

 the transverse and the latter the conjugate diameter. Taking these 

 as axes, and calling the halves of them respectively Jl and B, the 

 equation of the ellipse becomes Ji z y z -f B* x* = A* B*', in which JL 

 and B remain constant for the same ellipse. The points F and F', 

 are called the foci of the ellipse : and the sum of their distances FP 

 and F' P, from any point on the curve, is a constant quantity, always 

 equal to the transverse diameter. In the hyperbola, Fig. 19, the 

 difference of the distances F' P and FP is constant; and in the para- 

 bola, Fig. 18, the point P is equidistant from the focus F, and the 

 directrix CD. The equation of the parabola, taking its vertex as 

 the origin of coordinates, is y = Px; and that of the hyperbola, 

 referred to the middle of its transverse diameter, is Jl*y z B z x z = 

 A* B*. The applications of these equations, we have no room to 

 explain. 



