Abstract
According to Prof. Sophus Lie, the "theory of Differential Equations is the most important branch of modern mathematics." During the last century, this branch of mathematical science has been developed in a number of different directions, one of the most important of which is that based on the theory of transformation groups. As is well known, this whole method was originated by Lie in 1869--70, when he showed that most of the older theories of integration owe their origin to a common source and at the same time introduced new theories of integration, based on the theory of groups. In order to apply Lie's method to the problem of integration, it is necessary to know what group, if any, a given differential equation admits of. In his "Vorlesungen uber Differentialgleichungen mit Bekannten Infinitesimalen Transformationen" and various other publications, particularly in Vol. XXXII of the Mathematische Annalen, Lie established in complete detail all differential invariants of every group in two variables, and showed how to reduce as far as possible the problem of integrating the differential equations invariant under such groups. It would seem most desirable to do the same thing, as far as possible, for groups in x, y, z. This great problem has been solved only for a few special cases: by Lie, for example, for the group of Euclidean movements and a few other special groups (see "Continuierliche Gruppen," Kap. 22); by Tresse for the G10 of conform transformations (see Comptes Rendus, 1892, Tom. 114); by Dr. G. Noth in a Leipzig thesis on the differential invariants of a certain G 10. One object of the present paper is to begin the solution of this general problem in a systematic manner, by establishing the desired results for all G3's in x, y, x. The problem before us divides itself naturally into three parts: (I) The establishment of the normal forms of the Gr( r < 5) in n-variables. (II) The establishment of the differential invariants of the G3's in x, y, z: (i) when y and z are each functions of x; (ii) when z is a function of x and y. (III) Applications of, and remarks on, the results obtained.
