250 BOOK OF MATHEMATICAL PROBLEMS. 



1 202. Two surfaces have complete contact of the n A order at a 

 point : prove that there are n + I directions of normal sections for 

 which the curves of section will have contact of the n + 1 th order : 

 and hence prove that two conicoids which have double contact 

 with each other will intersect in plane curves. 



1203. Prove that it is in general possible to determine a 

 paraboloid whose principal sections shall be equal parabolas, and 

 which shall have a complete contact of the second order with 

 a given surface at a given point. 



1204. Prove that a paraboloid can in general be drawn 

 having a complete contact of the second order with a given surface 

 at a given point, and such that all normal sections through the 

 point have contact of the third order. 



1205. A skew surface is capable of generation in two ways by 

 straight lines ; at any point of it the absolute magnitudes of the 

 principal radii of curvature are a, b : prove that the angle be- 

 tween the generators which intersect in the point is 



_, a~ b 

 cos J r. 

 a + b 



1206. The points on the surface 



a (yz + zx + xy), 



at which the principal radii of curvature are equal and opposite, 

 lie on the cone 



and on the surface 



xyz = a*(x + y + z) 

 all such points lie on the cone 



x*(y + z) + y*(z + x) + z* (x + y) = 0. 



