64 



§ 10. All arbitiurj plane jr is a plane of sjminelry of one 0* 

 through four given points Ai; for through Ai there passes a pencil 

 of W's touching y* at its points of intersection with jt ; generally 

 one of these 0*'s passes through the mirror image of one of the 

 points Ai relative to ji and this surface satisfies the conditions. 



It may hapj)eii that the mirror image in question lies on the base 

 curve of the pencil; then :Tr is a plane of symmetry of all indivi- 

 duals of the pencil. As the sphere B tliBOugli Ai belongs to the 

 pencil, rr- must pass in this case through the centre J/ of this sphere. 



§ 11. The 00* planes of symmetry of the W's through /?w points 

 envelop a surface of which I shall determine the class. The 0''s the 

 |)lanes of symmetry of which pass through a point P of V^, cut 

 1^ along conies that touch y* at its points of intersection with a 

 ray of the plane pencil round P. The image of all such conies in 

 R^ is a quadratic cone K (0*'s through 4 points ^ 3). 



The quadratic surfaces through the 5 given points cut into V^ 

 a (/t^)^. that has an R^ as image in R^; this R^ cuts K along a 

 conic k^j/. 



To the degenerate conies of V^ there corresponds in R^ a cubic 

 hijpersurface, F\, that has a double surface 0\ of the 4''' order 

 (a surface of Veronese). Besides its two points of intersection with 

 k^'i (0"s through 4 points, § 3) that lie on ()\, k^j/ has 2 more 

 points in common with V^^, hence to the (^''''s through the 5 given 

 points the planes of symmetry of which pass through 7^, there belong 

 two paraboloids of revolution; these have F^ as a plane of sym- 

 metry. Through a ray p of the plane pencil round P there passes 

 one moj-e plane of symmetry that does not coincide with V ^, 

 consequently the planes of symmetry through P envelop a cone 

 that has P for vertex and that touches F , twice. An arbitrary 

 straight line / thiough P bears therefore 3 planes of symmetry; 

 through a line of V ^ there passes besides F^ only one more plane 

 of symmetry, hence: 



the planes of symmetry of the 0^'s through 5 given points envelop 

 a surface of the 3''' class of which V ^ is a double- tang ent plane. 



§ 12. The conic along which this surface touches F"^, has six 

 tangents that are the bearers of pencils of tangent planes; these 

 cannot belong to difFerent 0*'s for in that case through the 5 given 

 points there would pass a pencil of 0*'s touching y' at its points 

 of intersection with a straight line /; and from this w^ould follow 

 that the 5 given points must lie on a sphere. 



