Transformations of Surfaces Applicable to a Quadric

Luther Pfahler Eisenhart
1919 Transactions of the American Mathematical Society  
If a conjugate system of curves, or net, N, on a surface S and a congruence G of straight lines are so related that the developables of G meet S in N, the net and congruence are said to be conjugate. Two nets conjugate to the same congruence are said to be in the relation of a transformation T, if the nets are not parallel. In a previous paperf the author developed a general theory of these transformations T. When two surfaces, S and S, are applicable, there is a unique net on S which is
more » ... n S which is deformed into a net on S. Let N and N denote these nets. Petersonif showed that if a net N' parallel to N is known, a net N' parallel to N can be found by quadratures such that N' and N' are applicable. In a former paper § the author showed that when two such parallel nets N' and N' are known, two new applicable nets Ni and Ni can be found by a quadrature such that Ni and Ni are T transforms of N and N respectively. In the present paper these general results are applied to the case where N is a net on a quadric, that is when N lies upon a surface applicable to a quadric. We consider first the case where the quadric is a general central quadric not of revolution and find readily the T transforms of N into nets of the same kind as described in the following theorems: Theorem A. If N is a net applicable to a central quadric, not of revolution, Q, there can be found by the solution of a Riccati equation and quadratures three sets of co2 T transforms Ni which are applicable to Q; these transforms are conjugate to cc1 congruences G, there being co1 transforms conjugate to each congruence G; the lines of these congruences G through a point of N form a quadric cone; the tangent planes at points of a line of G to the nets Ni conjugate to it envelope a quadric cone and the points on Q corresponding to these points of the nets N\ on a line of G lie on a conic. Theorem B. If N is a net applicable to Q, there exist an infinity of sets
doi:10.2307/1988805 fatcat:nahjvzc5mbcbldjf2fegavstfm