  The problems in variation here concerned are such as to admit a continuous
group (in Lie's sense); the conclusions that emerge from the corresponding
differential equations find their most general expression in the theorems
formulated in Section 1 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 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.
