There are many different standpoints from which we might choose, out of the totality of all conceivable functions, extensive classes worthy of a particularly thorough investigation. Consider, for [Pg 32] example, the class of functions characterized by ordinary or partial algebraic differential equations. It should be observed that this class does not contain the functions that arise in number theory and whose investigation is of the greatest importance. For example, the before-mentioned function satisfies no algebraic differential equation, as is easily seen with the help of the well-known relation between and , if one refers to the theorem proved by Holder,[46] that the function satisfies no algebraic differential equation. Again, the function of the two variables and defined by the infinite series which stands in close relation with the function , probably satisfies no algebraic partial differential equation. In the investigation of this question the functional equation will have to be used. If, on the other hand, we are lead by arithmetical or geometrical reasons to consider the class of all those functions which are continuous and indefinitely differentiable, we should be obliged in its investigation to dispense with that pliant instrument, the power series, and with the circumstance that the function is fully determined by the assignment of values in any region, however small. While, therefore, the former limitation of the field of functions was too narrow, the latter seems to me too wide. The idea of the analytic function on the other hand includes the whole wealth of functions most important to science, whether they have their origin in number theory, in the theory of differential equations or of algebraic functional equations, whether they arise in geometry or in mathematical physics; and, therefore, in the entire realm of functions, the analytic function justly holds undisputed supremacy. [41] Crelle's Journal, vol. 84 (1878), and Atti d. Reale Acad. di Napoli, 1880. [42] Leipzig, 1897. Cf. especially Abschnitt I, Chaplets 2 and 3. [43] Symmetrie der regelmässigen Systeme von Figuren, 1890. [44] Krystallsysteme und Krystallstruktur, Leipzig, 1891. [45] Math. Annalen, vol. 53. [46] Math. Annalen, vol. 28. 19. ARE THE SOLUTIONS OF REGULAR PROBLEMS IN THE CALCULUS OF VARIATIONS ALWAYS NECESSARILY ANALYTIC? One of the most remarkable facts in the elements of the theory of analytic functions appears to me to be this: That there exist partial differential equations whose integrals are all of necessity analytic [Pg 33] functions of the independent variables, that is, in short, equations susceptible of none but analytic solutions. The best known partial differential equations of this kind are the potential equation and certain linear differential equations investigated by Picard;[47] also the equation