23G BOOK OF MATHEMATICAL PROBLEMS. 



1 1 29. An ellipsoid is generated by the motion of a point fixed 

 in a certain straight line, which moves so that three other fixed 

 points on it lie in the co-ordinate planes: prove that there are four 

 such systems of points; and that if the corresponding straight 

 lines be drawn through any point on the ellipsoid, the angle 

 between any two is equal to that between the other two. 



1130. Of two equal circles, one is fixed and the other moves 

 parallel to a given plane and intersects the former in two points : 

 prove that the locus of the moving circle is two elliptic cylin- 

 ders. 



1131. At each point of a generating line of a conicoid is 

 drawn a straight line touching the conicoid and at right angles 

 to the generating line : prove that the locus of such straight lines 

 is a hyperbolic paraboloid whose principal sections are equal 

 parabolas. 



1132. The locus of the axes of sections of the surface 



ax* + by* + cz* = 1, 

 which contain the line 



? = H = - 

 I m n ' 



is the cone 



(b - c) yz (mz - ny) + (c-a) zx (nx - Iz) +(a-b) xy (ly - mx) = 0. 



1133. The three acute angles made by any system of equal 

 conjugate diameters of an ellipsoid will be together equal to two 

 right angles, if 



2 (2o f - b> - c 2 ) (W -c 2 - a') (2c 2 - a 3 - I 2 ) = 27o W ; 

 2a, 2b, 2c being the axes. 



1134. From different points of the straight line 



xz 



