TWENTY-SIXTH ANNUAL MEETING. 25 



sion, the circular sections are at right angles: if they are in geometrical progres- 

 sion the tangent planes at the umbilici touch the sphere through the central 

 circular sections. 



2. If the squares of the axes of a pedal surface of the "first kind " are in 

 harmonical progression, the umbilici lie on the central circular sections: if they 

 are in arithmetical progression, the two systems of spheres cutting the circular 

 sections have their "central diameters" at right angles: if they are in geomet- 

 rical progression, the central tangent sphered at the umbilici touch the sphere 

 through the central circular sections. 



1. The angle made by the two systems of planes cutting circular sections 

 from an ellipsoid is k = 2tan^— f^lil£i!l ^. 



2. The angle between the "central diameters" of the two systems of cen- 

 tral spheres cutting circular sections from a pedal surface of the "first kind " is 



Since a straight line in space inverts into a circle through^ the origin, the 

 straight-line generators of a conicoid invert into circle generators of a pedal sur- 

 face. Hence the properties of ruled conicoids, when inverted, give rise to cor- 

 responding properties of circularly ruled pedal surfaces. This circular generator 

 lies entirely in the surface and passes through the origin. And since the hyper- 

 boloid of one sheet is the only central conicoid having straight-line generators, 

 the pedal surface of the " second kind " is the only one of the three surfaces un- 

 der discussion having circular generators. Also, corresponding to the two sys- 

 tems of straight-line generators of the hyperboloid are two systems of circular 

 generators of the pedal surface. Following are a few theorems on the pedal 

 surface of the "second kind" obtained by inverting those properties of « the 

 hyperboloid of one sheet which relate to straight-line generators: 



1. The tangent plane to an hyperboloid of one sheet at a point through which 

 a generating line passes will contain that generating line. 



2. The central tangent sphere to a pedal surface of the "second kind " at a 

 point through which a generating circle passes will contain that circle. 



1. Through any point on a generating line of an hyperboloid of one sheet an- 

 other generating line passes, and they are both in the tangent plane at the point. 



2. Through any point on a generating circle of a pedal surface of the " second 

 kind" another generating circle passes, and they are both in the central tangent 

 sphere at the point. 



1. Any plane through a generating line of a conicoid touches the surface, its 

 point of contact being the point of intersection of the two generating lines which 

 lie upon it. 



2, Any central sphere through a generating circle of a pedal surface touches 

 the surface, its point of contact being the point of intersection of the two gener- 

 ating circles which lie upon it. 



1. Three non-intersecting generators are suflBcient to determine the conicoid 

 on which they lie. 



2. Three non-intersecting (except, of course, at the origin) circular generators 

 are suflBcient to determine the pedal surface on which they lie. 



1. The straight lines which intersect three fixed non-intersecting straight 

 lines are generators of the same system of a conicoid, and the three fixed lines are 

 generators of the opposite system of the same conicoid. 



