DIFFERENTIAL EQUATIONS OF THE TYPE : y " ' = Gy " +

Kasner, "Transformation Theory of Isothermal Families and Certain Related Trajectories," Revista de Matematica, 2, 17-24 (1941). De Cicco, "The Two Conformal Covariants of a Field," Revista de Matematica, 2, 59-66 (1941). 2 A generalization of Kasner's theorem has already been given in another direction. This other work is concerned with field-element to lineal-element transformations. See Kasner and De Cicco, "Generalized Transformation Theory of Isothermal and Dual Families," Proc. Nat. Acad. Sci., 28, 52-55 (1942). ' The word wex was introduced by Kasner to denote any doubly-infinite system of plane curves. It is therefore represented by a general differential equation of second order y' = f(x, y, y'). 4 See Kasner, Princeton Colloquium Lectures, 1912, 1934. Also Kasner, "Isothermal Systems in Dynamics," Bull. Am. Math. Soc., 14, 169-172 (1908); and "A Characteristic Property of Isothermal Systems of Curves," Mathematische Annalen, 59, 352-354 (1904). 6 The content of our new group is expressed by the symbol 2 O 6f(1), whereas the content of Kasner's group, equations (1), is 2 o 1 + 4f(1). See Kasner, "A Notation for Infinite Manifolds," Am. Math. Monthly, 49, 243-244 (1942).