Important autograph manuscript in which Appell generalises the theory of Abelian functions, due principally to Abel, Jacobi, Riemann and Weierstrass, to a class of functions he terms ‘fonctions à multiplicateurs’, and investigates their integrals and Fourier expansions. “His scientific work consists of a series of brilliant solutions of particular problems, some of the greatest difficulty. He was a technician who used the classical methods of his time to answer open questions, work out details, and make natural extensions in the mainstream of the late nineteenth century” (DSB), which cites the present paper as one of his most notable works. Appell’s memoir won honorary second prize in the competition mentioned on the title, the first prize going to Henri Poincaré for his great work ‘Sur le problème de trois corps et les équations de la dynamique.’ In his statement on behalf of the prize committee, Charles Hermite wrote about Appell’s contribution: “Le mémoire de M. Appell, sur les intégrales de fonctions à multiplicateurs et leur application au développement des fonctions abéliennes en séries trigonométriques, est également digne du plus haut intérêt. M. Appell a ouvert un champ nouveau dans la théorie des fonctions d’une variable, en donnant l’origine d’une catégorie de transcendants, douées de propriétés extrêmement remarquables, dont il a fait une étude approfondie et qui sont appelées à jouer un grand rôle. C’est, à notre époque, un des plus importants résultats de l’Analyse que ces découvertes de nouvelles fonctions auxquelles s’attachent les noms illustres de’Abel et de Jacobi, de Göpel, de Rosenhaim, de Weierstrass et Riemann. M. Appell s’est surtout inspiré de Riemann; son beau Mémoire, ceux de M. Poincaré sur les fonctions fuchsiennes, d’autres travaux français continuent l’oeuvre de ces grands géomètres.” No manuscripts by Appell are listed on ABPC.
“In 1885 Gösta Mittag-Leffler could look back at a successful start on his career as the first professor in mathematics of the newly founded Stockholms högskola (later to become Stockholm University). After his education and doctor’s degree from Uppsala, he had spent three years in Paris and Berlin while he forged firm bonds with many leading European mathematicians. He then turned to Helsinki where he broadened his horizons as a dedicated and strong-willed professor. Four years later, in 1880, he returned to his native Stockholm. The new principles on which Stockholms högskola was founded gave Mittag-Leffler the opportunity to work out his ideas of how mathematical education and research should be organized and promoted. He attracted several talented postgraduates, and managed to engage Sofia Kovalevskaya as lecturer and later full professor at the department. The journal Acta Mathematica which he founded in 1882 was right from the beginning received as a leading international publication. It established itself as an effective transmitter of new mathematical ideas between France and Germany in the aftermath of the Franco-Prussian War (1870-71).
“One of Mittag-Leffler’s inventive ideas in promoting Acta Mathematica was to turn to King Oscar II of Sweden and Norway, both for financial support of the project and as the first enlisted subscriber. In 1884 another grand idea from Mittag-Leffler had matured. Again involving King Oscar, he now wanted to arrange an international prize competition in mathematics honouring the 60th birthday of the king. It seems that Mittag-Leffler first revealed his plan to Kovalevskaya. We find a short reference in a letter to her from 4 May 1884. A month later it is clear from another letter by Mittag-Leffler that she had by then discussed the prize with Karl Weierstrass. In July Mittag-Leffler for the first time briefed Weierstrass on the plans.
“It was first decided to form a small prize jury consisting only of three members: Mittag-Leffler himself, acting as administrative and coordinative liaison with his mentors and friends Karl Weierstrass in Berlin and Charles Hermite in Paris. They were not only the two dominant mathematicians of the older generation, but there was also a special sympathy between them. This would be a prize awarded not for past contributions, but for a solution to an unsolved problem specified by the committee. In order to attract the best mathematicians from different branches of mathematical analysis they agreed on four questions. The first three seem to have been proposed by Weierstrass and the fourth by Hermite. It took a year of intense discussions to settle all the details, which were then officially announced in Volume 7 of Acta Mathematica in the middle of 1885.
“The special features of this competition was the international and ambitious appeal, and the connection not to an academy or institution, but to the journal Acta Mathematica, where the winning entry finally was to be published. The prize consisted of a gold medal and 2,500 Swedish kronor. (As a comparison, Mittag-Leffler’s annual salary as professor was 7,000 kronor.) The memoirs should be submitted before 1 June 1888 (almost three years after the announcement), with anonymity maintained through a motto on an enclosed sealed envelope containing the name of the author.
“The first question was a formulation of the classical n-body problem in celestial mechanics, and it attracted the most attention: “For a system of arbitrarily many mass points that attract each other according to Newton’s law, assuming that no two points ever collide, find a series expansion of the coordinates of each point in known functions of time converging uniformly for any period of time.” Weierstrass nourished great hopes that there would be a method of solution based on a simple fundamental idea, and that such a solution could help conclude whether the system was stable. In applying this formulation of the n-body problem as a model of the solar system, could it be excluded that the planetary orbits might alter radically over a long time, or even that one planet be thrown out of the system? …
“A list of all the twelve manuscripts received by June 1888 was published in Volume 11 of Acta with their identifying epigraphs. It turned out that five of the authors had attempted the prestigious Question 1, one had tried Question 3, while six treated a subject of their own, which remained a secondary option. Only four of the authors have been identified: in addition to Poincaré also Paul Appell, Guy de Longchamps and Jean Escary … Although no entry had actually solved any of the questions, Mittag-Leffler and his jury were soon of the preliminary opinion that Poincaré was in a class of his own, that Appell should be awarded a second honorary prize, and that no other entries needed much further examination” (Rågstedt).
In 1873, Appell (1855-1930) entered the École Normale, from which he graduated first in the class of 1876, three months after earning his doctorate. From this time on, Appell was highly active in teaching, research, editing, and public service. He typically held several teaching posts at the same time, including the chair of mechanics at the Sorbonne from 1885. He was elected to the Académie des Sciences in 1892. He served as dean of the Faculty of Science of the University of Paris from 1903 to 1920 and as rector from 1920 to 1925. In various government posts, including membership in the Conseil Supérieure d’Instruction Publique, he was an exponent of educational reform and initiator of numerous large-scale projects, including the Cité Universitaire. He vigorously supported the movement for women’s rights, carried from Alsace his brother’s reports destined for the French War Office, defended his fellow Alsatian Dreyfus and served on an expert commission whose ruling played a key role in his final rehabilitation. During World War I he founded and led the Secours National, a semi-official organization uniting all religious and political groups to aid civilian victims. He described the return of the tricolour to Alsace as the fulfilment of his “lifelong goal” and felt that Germany had been treated too easily. He served as secretary-general of the French Association for the League of Nations.
“Appell’s first paper (1876) was his thesis on projective geometry in the tradition of Chasles, but at the suggestion of his teachers he turned to algebraic functions, differential equations, and complex analysis. He generalized many classical results (e. g., the theories of elliptic and of hypergeometric functions) to the case of two or more variables. From the first his work was close to physical ideas. For example, in 1878 he noted the physical significance of the imaginary period of elliptic functions in the solution of the pendulum problem, and thus showed that double periodicity follows from physical considerations. In 1880 he wrote on a sequence of functions (now called the Appell polynomials) satisfying the condition that the derivative of the nth function is n times the previous one.
“In 1885 Appell was awarded half the Bordin Prize for solving the problem of “cutting and filling” (deblais et remblais) originally posed by Monge: To move a given region into another of equal volume so as to minimize the integral of the element of volume times the distance between its old and new positions. In 1889 he won second place (after Poincaré) in a competition sponsored by King Oscar II of Sweden: To find an effective method of calculating the Fourier coefficients in the expansion of quadruply periodic functions of two complex variables.
“The flow of papers continued, augmented by treatises, textbooks, and popularizations and seemingly unaffected by other responsibilities. Although Appell never lost his interest in “pure” analysis and geometry, his activity continued to shift toward mechanics, and in 1893 Volume 1 of the monumental Traité de mécanique rationnelle appeared. Volume V (1921) included the mathematics required for relativity, but the treatise is essentially an exposition of classical mechanics of the late nineteenth century. It contains many of Appell’s contributions, including his equations of motion valid for both holonomic and non-holonomic systems, which have not displaced the classical Lagrangian system in spite of undoubted advantages” (DSB).
Rågstedt, ‘From order to chaos: the prize competition in honour of King Oscar II,’ mittag-leffler.se/library/prize-competition
"Sur les intégrales de fonctions a multiplicateurs et leur application au développment des fonctions abéliennes en séries triginométriques" + "Supplémet". Autograph manuscript in 8vo (220 x 175 mm), 278 leaves written on recto page, consisting of title (1 leaf), device leaf (1 leaf), Table des matières (1 leaf) and headtext leaf 1-210 (including 4bis, 156 a-d, 171 bis) + Supplément title (1 leaf) and headtext 1-60 and Table des Matières de supplément (4 leafs). Illustrations in text. Partly worn, some marginal tears. Custom cloth box.