APPENDIX 



II : ... from equations (1), (2), and < 



^ = 8 -i^ = ,aconstant, (4) 



PL MII a 



i.e.,the r;iti. /'/ : /'A. i-.r every point P of the section >/'/ 



until Tun 1. ArtN.-K 17.~>) the section is a second degree curve, \\ith a focub 



at /'.directrix <;/>L, and eccentricity i!ll 



Mil ((. 



Similar the other focus, and the line of intersection of the 



11 K a il .!'/:,"/$' is the other directrix of the conic S7'A'(^; hence 

 the thec uvni is established. 



.Moreover, the plane VW, being perpendicular to the axis of the < 

 and u\"\\\ being a section made by a plane passing through the axis, 

 a = Z0nr, and is constant for a given cone, while $ = Z OSR, an<l 

 vari.-s only \\itli tlie plane UK. 



Il.-ncv the eccentricity varies with the inclination of the plane UK. 

 and there are the three following cases : 



if 6 < a, then e < 1, and the section is an ellipse; 

 if = , then e = 1, and the section is a parabola; 

 if 6 > a, then e > 1, and the section is an hyperbola. 



Again, if the cutting plane HK passes through the vertex O of the 

 cone, then the focus F is on the directrix GDL, and the section will be 

 either a pair of straight lines or a point : 



if $ < a, the section is a point, the vertex O of the cone. 



if $ = a, the section is a pair of coincident straight lines, an element of 

 the cone ; 



if $> Uj the section is a pair of intersecting straight lines, two elements 

 through the vertex (cf. Note C). 



It is, of course, evident that for every elliptic section of the i 

 spheres both lie in the same nappe of the cone, and touch the plane of 

 the section (UK) on opposite sides; while for every hyperbolic H 

 these focal spheres lie one in each nappe of the cone, and both on the 

 same side of the plane of the section. 



In the above proof, for the sake of simplicity, a right circular cone 

 was employed; it is easy to show (see Salmon's Conic Sections, j>. 

 that every section of a second degree cone (right or oblique) by a plane 

 is a second degree curve. 



