On the other hand I wish to emphasize the fact that there certainly exist analytical functional equations whose sole solutions are non-differentiable functions. For example a uniform continuous non-differentiable function can be constructed which represents the only solution of the two functional equations where and are two real numbers, and denotes, for all the real values of , a regular analytic uniform function. Such functions are obtained in the simplest manner by means of trigonometrical series by a process similar to that used by Borel (according to a recent announcement of Picard)[13] for the construction of a doubly periodic, non-analytic solution of a certain analytic partial differential equation. [Pg 18] [10] Lie-Engel, Theorie der Transformationsgruppen, vol. 3, Leipzig, 1893, §§ 82, 144. [11] "Ueber den analytischen Charakter der eine endliche Kontinuierliche Transformationsgruppen darstellenden Funktionen," Math. Annalen, vol. 41. [12] Werke, vol. 1, pp. 1, 61, 389. [13] "Quelques théories fondamentales dans l'analyse mathématique," Conférences faites à Clark University, Revue générale des Sciences, 1900, p. 22. 6. MATHEMATICAL TREATMENT OF THE AXIOMS OF PHYSICS. The investigations on the foundations of geometry suggest the problem: To treat in the tame manner, by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theory of probabilities and mechanics. As to the axioms of the theory of probabilities,[14] it seems to me desirable that their logical investigation should be accompanied by a rigorous and satisfactory development of the method of mean values in mathematical physics, and in particular in the kinetic theory of gases. Important investigations by physicists on the foundations of mechanics are at hand; I refer to the writings of Mach,[15] Hertz,[16] Boltzmann[17] and Volkmann.[18] It is therefore very desirable that the discussion of the foundations of mechanics be taken up by mathematicians also. Thus Boltzmann's work on the principles of mechanics suggests the problem of developing mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua. Conversely one might try to derive the laws of the motion of rigid bodies by a limiting process from a system of axioms depending upon the idea of continuously varying conditions of a material filling all space continuously, these conditions being defined by parameters. For the question as to the equivalence of different systems of axioms is always of great theoretical interest.