328 



WILSON AND MOORE. 



of M is the pedal of the indicatrix and hence we have the theorem: 

 The inverse of the pedal of the indicatrix is the locus of points where con- 



secidive normal planes 

 about a point intersect the 

 normal plane at the point}'^ 

 Consider next the case 

 n = 5. Here the indica- 

 trix is a conic which may 

 or may not he in a plane 

 through 0. In the latter 

 special case the reasoning 

 before holds except for the 

 fact that the solution for r 

 in (92') is no longer a point, 

 but a line through that 

 point perpendicular to the 

 plane of the conic. The 

 locus of intersection of 

 consecutive normal spaces 

 is therefore a right cylin- 

 der of which the directrix 

 is the conic which is the inverse of the pedal of the indicatrix. This 

 is merely a direct extension of the case previously treated. 



The general case. If the indicatrix does not lie in a plane with 0, 

 and if we lay off along a the distance equal to the radius of curvature, 

 instead of equal to the curvature, we get a point Q which lies both 

 on the cone determined by as vertex and the indicatrix as directrix 

 and on the sphere through which is the inverse of the plane of the 

 indicatrix. The locus of Q is therefore a sphero-conic. The plane 

 r«a = 1 passes through the point Q and is perpendicular to a; it 

 therefore passes through the point 0' of the sphere diametrically 

 opposite to 0, this point 0' being also the inverse of the foot F of the 

 perpendicular OF from upon the plane of the indicatrix. 



Now r*Ji = is the plane through perpendicular to 1^, and hence 

 perpendicular to the plane of the indicatrix, and hence finally r • |JL = 

 is a plane through the line OF. The intersection of r«|A = andr-a 

 = 1 is therefore a line through 0' perpendicular alike to p- and a, and 

 consequently perpendicular to the plane tangent to the cone (described 

 by a) through the element a (since p. is parallel to the tangent to the 

 indicatrix at the extremity of a). 



O 

 Figure 2. 



37 Kommerell, loc. cit. 



