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- 

 secutive normal planes 

 about a point intersect the 

 normal plane at the point.^'^ 

 Consider next the ease 

 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 in^'erse 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 • Hi. = is the plane through perpendicular to p-, and hence 

 perpendicular to the plane of the indicatrix, and hence finally I'H- = 

 is a plane through the line OF. The intersection of r-^- = andr*a 

 = 1 is therefore a line tlirough 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 M- is parallel to the tangent to the 

 indicatrix at the extremity of a). 



Figure 2. 



37 Kommerell, loc. cit. 



