AbstractsMathematics

The study of continuous groups of transformations in function space was begun by G. Kowalewski in 1911. Vessiot considered the conditions under which r parameter Volterra transformations form a group. L. L. Dines considered projective transformations in function space continuing Kowalewski's work. I. A. Barnett gave a few examples of functionals invariant under one-parameter continuous groups of transformations in the space of continuous functions. The systematic study of functionals invariant under continuous groups of transformations in function space, and also the study of functionals involving derivatives of functions and invariant under groups of transformations in function space have not been considered previously, so far as the writer has been able to ascertain. Consequently the author commences the study of the latter functionals, i.e., functionals involving derivatives of functions (and as a special case, integro-differential equations), that are invariant under a Volterra one parameter group of continuous transformations, a problem allied to the calculus of variations.