> .1 N \LYTH CKtMKiliY [Cu IV. 



point nln-ay* bears a constant ratio to its distance 



fixed straight line. Tin-si- ourvefl are called tin- Conic 



Sections, or more briefly Conies, because they can be obtained 



lu* curves of intersection of planes and right circular 

 cones;* in fact, it was in this way that they first became 

 known. The last thive examples just given belong to tins 

 class, although it is only in No. 4 that this fact is directly 

 Stated. These loci are the parabola, the ellipse, and the 

 hyperbola; it will be .sli..\\n later that they include as spe- 

 cial cases the straight line and the circle. f They are of 

 primary importance in astronomy, where it is found that the 

 orbit of a heavenly body is a curve of this kind. 



The general equation, which includes all of these curves, 

 will now be derived, and the locus briefly discussed; in a 

 subsequent chapter will be given a detailed study of the 

 properties of these curves in their several special forms. 

 (a) The equation of the locus. Let F be the fixed point, 

 the focus of the curve ; D'D the fixed 

 line, the directrix of the curve ; and * 

 the given ratio, the eccentricity of 



the curve. 



/ The coordinate axes may of course 



- be chosen as is most convenient. Lei 



Fio.33. &D b the y-axis, and the perpendicu- 



lar to it through /'. />.. the line OFX, 

 be the x-axis. Let P = (a;, y) be an} r position of the generat- 

 ing point, and let OF, the fixed distance of the focus from 

 the direetrix.be denoted by/-; then the coordinates of the 

 foeiw are (, 0). Connect F and P, and through P draw 

 /,/' p'-rpendicular to the directrix'. 

 Then FP : LP = e, [geometric equation] 



* See Note D, Appendix. t See Note C, Appendix. 



