Some Theorems of Comparison and Oscillation

+ S&iX2(Xi — X2)} = 0. Clearly, the quartic degenerates into the #-line of the pencil of cubics, and into a cubic having (si, s2) sz) as the $-point. Hence the THEOREM: Any cubic of the net with a given S-point may be generated by any pencil of cubics within the net, not containing the given cubic, and the projective pencil of lines joining the Spoint of the given cubic to the S-points of the cubics of the pencil. Consider next two pencils of cubics Cx with the S-line I, and C^ with the /S-line m, and a point S, not on I or m. Draw any line g through S, cutting I and m in SA and SM, and construct the cubics Ck and C^ having #A and Sy as S-points. They both pass through the two fixed points P and P ' on g corresponding to each other in the Steinerian transformation. But P and P ' also lie on the cubic C8 associated with S as an /S-point. For a variable g through S, SA and fiy describe two perspective point sets on I and m which are projective with the pencils of cubics Ck and C^. These pencils are therefore themselves projective, and generate the cubic Cs. Hence the THEOREM: Every cubic of the net associated with a Steinerian transformation may be generated in an infinite number of ways by projective pencils of cubics of the same net