BOOK OF MATHEMATICAL PROBLEMS. 



IV. Tetraliedral Coordinates. 



1 1C2. When the opposite edges of the tetrahedron ABCD are, 

 two and two, at right angles, the three shortest distances between 

 the opposite edges meet in the point 



a (A& + AC* + AD* - *) = /3(BC* + BD* + BA* - k)= ... = ..., 

 k bi ing the sum of the squares on any pair of opposite edges. 



1163. Determine the condition that the straight line 



' . ' z 



X p v 

 may touch the conicoid 



I fly + mya + na/3 + I'aS + m'fiS + n'yb = ; 



and thence prove that the equation of the tangent plane, at the 

 point a = /? = y = 0, is 



l'a + m/3 + n'y = 0. 



1164. Any conicoid which touches seven of the planes 



la ra/? ny r8 = 



will touch the eighth ; and its centre will lie on the plane 

 Pa + m 2 /? + ri*y + r 2 8 = 0. 



Prove that this plane bisects the part of each edge of the fun- 

 damental tetrahedron which is intercepted by the given planes. 



1165. If a hyperbolic paraboloid be drawn containing the 

 sides A By BC, CD, DA of a quadrilateral which is not plane, 

 and /' be any point on this surface, 



vol. PBCD x vol. PDAS = vol. PD A C x vol. PABC. 

 Also, if any tangent plane meet AJ1, CD in P, Q respectively, 

 AP : BP :: DQ : CQ. 



