384 \rrENDix 



Again, suppose the focus F to be on the directrix. Then, it /' is any 

 point of the locus, and LP perpendicular to FD, 



n* = e-LP, (1) 



and 8inZPFL= = l; . 



hence the angle PFL is constant, with two supplementary values for a 



value of f. 



The locus consists therefore of two straight lines intersecting at F t 

 and equation (2) shows that : 



if e > 1, the lines are real and different ; 

 if e = 1, the lines are real and coincident; 



and if e<l, the lines are imaginary, and the real part of the locus 

 consists of the point F. 



Suppose now the directrix, with the focus upon it, to be at infinity; 

 then, if e > 1, the locus is a pair of parallel lines. 



These results agree with those already summarized in Art. 182. 



NOTE D 



Sections of a cone made by a plane. The following proposition is 

 due to Hamilton, Quetelet, and others (see Taylor's Ancient and Mod- 

 ern (Geometry of Conies, p. 204). 



If a right circular cone is cut by a plane, and two spheres are inscribed 

 in the cone and tangent to this plane, then the section of the cone made 

 by the plane is a second degree curve (cf. Part I, Arts. 48, 175), of which 

 the foci are the points of contact of the spheres and the plane, and ilu- 

 trices are the lines in which this plane intersects the planes of the 

 circles of contact of the spheres and the cone. 



CON- n:t ( i ION : Let O-VW be a right circular cone cut by the plane 

 UK in the section RPSQ, P being any point of the section. Inscribe 

 two spheres, C-ABF and C'-A'BF, whose circles of contact with the 

 cone are AEB and A'E'ff, respectively, and which are tangent to the 

 plane UK in the points F and P. Through P draw the element OP of 

 the cone, cutting the circles of contact in the points E and E'. Also 

 pass a plan*- M\ through the circle AEB, and therefore perpendicular 

 to the axis OCC of the cone ; it will intersect the plane HK in a straight 



