SOLID GEOMETLY. :M-1> 



1197. The envelope of all paraboloids, to whieli a <:' 

 tetrahedron is self-conjugate, is the planes each of which bisects 

 three edges of the tetrahedron. 



1198. A prolate spheroid can be described having two oppo- 

 site umbilici of an ellipsoid as foci and touching the ellipsoid along 

 a plane curve: and this spheroid will be the envelope of a series 

 of spheres, having one system of circular sections of the ellipsoid 

 as great circles. 



1199. Spheres are described on a series of parallel chords of 

 an ellipsoid as diameters : prove that they will have double con- 

 tact with another ellipsoid ; and that the focal ellipse of the latter 

 will be the diametral section of the former conjugate to the 

 chords. Also, if a, 6, c be the axes of the former, and a, /?, y of 

 the latter, 



y bring that axis which is perpendicular to the plane bisecting 

 the chords. 



1200. The envelope of a sphere, intersecting a given conicoid 

 in two planes and passing through the centre, is a surface of the 

 fourth degree, touching the conicoid along a spherical conic. 



VIII. Curvature. 



1201. If from any point of a curve equal small lengths St be 

 measured in the same direction along the curve, and along the 

 circle of absolute curvature, respectively ; the distance between 

 the extremities of these lengths tely 



/-. <r being the radii of curvature and torsion respectively at tlu- 



