The problems in variation here concerned are such as to admit a continuous group (in Lie's sense); the conclusions that emerge for the corresponding differential equations find their most general expression in the theorems formulated in Section I and proved in following sections. Concerning these differential equations that arise from problems of variation, far more precise statements can be made than about arbitrary differential equations admitting of a group, which are the subject of Lie's researches. What is to follow, therefore, represents a combination of the methods of the formal calculus of variations with those of Lie's group theory. For special groups and problems in variation, this combination of methods is not new; I may cite Hamel and Herglotz for special finite groups, Lorentz and his pupils (for instance Fokker, Weyl and Klein)¹ for special infinite groups. Especially Klein's second Note and the present developments have been mutually influenced by each other, in which regard I may refer to the concluding remarks of Klein's Note.