1. D. D. Kosambi,The existence of a metric and the inverse variational problem, Bull. U. P. Acad. of Science, vol. 2. The main ideas of the investigation were set forth in a lecture to the Aligarh Mathematical Seminar on March 5. 1931. Some of the results of the present paper have also been given in a note in the Rendiconti R. Accad. dei Lincei16 (1932), S. 410?415.
2. In a personal letter, an extract of which is published after this paper.
3. D. R. Davis in the Bull. Amer. Math. Soc. (1929), pp. 371?380. The equations given there would seem to be necessary, but not sufficient.
4. For a detailed bibliography of the subject, and in particular for references to the numerous papers of Berwald, I refer the reader to the article of Koschmieder, Jahresber, d. D. M. V.40, pp. 109?132. Other papers related to the present investigation are D. R. Davis, l. c. and Trans. Amer. Math. Soc. 33, p. 246 and J. Douglas, Ann. of Math. (II)29.
5. See Eisenhart,Non-Riemannian Geometry, Am. Math. Soc. Coll. (1927), pp. 64?67.