Lineal Element Transformations Which Preserve the Dual-Isothermal Character.

THEOREM. In order that a system S of o I curves represents the additivemultiplicative trajectories ofa given field, it is necessary (but not sufficient) that S possesses the dynamical Property I; this may be stated simply in either of these alternative geometric forms: Property Ia: For the o I curves of S passing through any lineal element E of the plane, construct the osculating parabolas at E. The locus of the foci of these is a circle passing through the point of E. Property Ib: The 1 directrices of these osculating parabolas all pass through afixed point, thusforming a pencil of straight lines. Differential equations of the general form (10), that is, triply infinite systems of curves possessing Property I, have presented themselves in a great variety of physical and geometrical problems. For example, dynamical trajectories, catenaries, brachistochrones, sectional families and curvature trajectories.2