SOLID GEOMETRY. 



11 GO. The locus of the centres of all conicoids which have 

 four common generators, two of each system, is a straight line. 



11 57. Perpendicalani are let fall from the point (a, /?, y, 8) 

 on the faces of the fundamental tetrahedron, and the feet of these 

 perpendiculars lie in one plane; prove that 



1111 



*,'%.v' l "vV ' 



Pi Pf P& P 4 being the perpendiculars of the tetrahedron. 



11G8. If a tetrahedron be self-conjugate to a given sph 

 any two opposite edges will be at right angles to each other, and 

 all the plane angles at one of the solid angles will be obtuse. 



1 1 69. If the opposite edges of a tetrahedron be, two and two, 

 _ht angles to each other; the circumscribed sphere, the 

 spin-re bisecting the edges, and the sphere to which the tetra- 

 hedron is self-conjugate will have a common radieal plane. 



1 170. A tetrahedron is such that a sphere can be described 

 touehing its six edges; prove that any two of the four tangent 

 cones drawn to this sphere from the angular points will have a 

 common tangent plane and a common plane section; and the 

 planes of these common sections will all six inert in a point. 



1171. A tetrahedron is such that the straight lines 

 its angular points to the points of contact of tin inserted .sphere 

 with tin- n-jM-ctivrly opposite faces nie<-t in a point ; prove tha f . 

 iet, the sides of the face on which the point 

 lies subtend equal angles. 



117:!. It" a 'Mnserihe a tetrahedron .1 HCD. and 



it planes at A y /?, C, /> form a tetrahedron A'BC'I?', 

 th.-., if .1.1", A'/;' intersect, CV", />// will ftlao intersect 



117 its are taken on a ,,,,!. oid ; pr..\e that, if the 



straight line joining one of the points to the pole of tin- plane 



1G 2 



