On certain loci of lines incident with curves and surfaces in four-space

In this paper is presented a partial list of formulas giving the orders of the ruled hypersurfaces and ruled surfaces whose rulings are incident with given curves and surfaces in 4-space. If the number of incidences is equivalent to six simple conditions, the number of lines having these incidences is finite. The formulas herein presented are analogous to those giving the orders of the surfaces whose rulings are lines satisfying three conditions and the number of lines satisfying four conditions in 3-space.* Some of these formulas are obvious, others require proof. Those that are not herein included are left out for future consideration. We shall let h be the number of apparent double points, p the deficiency, of a given curve C; H the number of apparent double points, P the deficiency, of a 3-space section c of a given surface F; and t the number of apparent triple points of F or the number of lines through a given point meeting F three times, f First consider the hypersurface F" whose lines satisfy four conditions. For a line to meet a given curve once is equivalent to two conditions. Hence the °o lines cutting across two given curves Cs C* form a hypersurface whose order is obviously