of a Conic Section. _ 189 



by which the focus was determined in Prop. I, it is evident 

 that the plane VAS must be perpendicular to 'the plane APQ, 

 and that the circle which touches the three straight lines J A, 

 VB, AS, must touch the last in the point S. Consequently, 



1. If the section be an ellipse (fig. i.) having the axis 

 AA', the difference of VA and J'A' will be equal to the dif- 

 ference of EA and BA', that is, to the difference of SA and 

 SA ; and therefore the locus in question will be an hyperbola 

 having the foci A, A', and. the principal vertices S, S'. 



2. If the section be a parabola (fig. 2.) Af^ will be equal 

 to the sum of VB and AS, that is, (if SG be taken equal to 

 AS, and if GK be drawn perpendicular to GA to cut VB pro- 

 duced in A',) equal to the sum of VB and BK, or to FK; and 

 therefore the locus in question will be a parabola equal to the 

 given one, having the focus A and the principal vertex ^S*. 



3. If the section be an hyperbola (fig. 3.) having the axis 

 AA, the sum of J'^A and FA' will be equal to the sum of EA 

 and BA', that is, to the sum of SA and SA; and therefore 

 the locus in question will be an ellipse having the foci A, A, 

 and the principal vertices S, S'. 



In every case, therefore, the locus is a conic section, which 

 pas.ses through the focus or foci of the given conic section, in 

 a plane at right angles to the plane of the latter, and has for 

 its focus or foci the principal vertex or vertices of the latter. 



Q. E. I. 



C(A. If two conic sections, having their planes at right 

 angles to one another, jjass each of tliem through the focus or 

 foci of the other, each of them shall be the locus of the vertices 

 of all right cones which have the other for a .section. 



Thus it appears that the focus is the point in which the 

 plane of the conic section is cut by the locus of the vertices 

 of all right cones having it for a section. In fact, it is not 



