MIAI'TER VIII 

 THE CONIC SECTIONS 



101. In Art. 4H, which should now IK- carefully i 

 a conic section was defined , its general equation was de- 

 rived ; its three species, viz., the parabola, ellipse, and hyper- 

 bola, were mentioned ; and a brief discussion of the nature 

 and forms of the curve was given. In the present chap- 

 ter, each of these three species will be examined somewhat 

 more closely than was done in Chapter IV, and some general 

 theorems concerning its tangents, normals, diameters, chords 

 of contact, and polars will be proved. 



The general equation (Art. 48) of the conic section 

 might here be assumed, and the special forms for the parab- 

 ola, the ellipse, and the hyperbola be derived from it; but, 

 partly as an exercise, and partly for the sake of freedom 

 to choose the axes in the most advantageous ways, the equa- 

 tions will here be re-derived, as they are needed, from the 

 definitions of the curves. 



I. Till. 1'AKAHOLA 

 Special Equation of Second Degree 

 + 2.FV + C = 0, or By* + ZGx + 2 Fy + C = 



102. The parabola defined. A parabola is the locus of 

 a point which moves so that its distance from a fixed point, 

 called the focus, is equal to its distance from a fixed line, 



170 



