## Thèses présentées a la Faculté des sciences de Paris pour obtenir le grade de docteur des sciences mathématiques. 1re These. Sur les propriétés des fonctions définies par les équations aux différences partielles. 2e These. Propositions données par la Faculté. Soutenues le 1er aout 1879, devant la commission d'examen.

Paris: Gauthier-Villars, 1879.

First edition, extremely rare, of Poincaré’s thesis for his doctorate in science from the University of Paris. “Poincaré’s doctoral thesis [was] on differential equations (not on methods of solution, but on existence theorems), which led to one of his most celebrated contributions to mathematics-the properties of automorphic functions” (Boyer & Merzbach, p. 675). “The theory of differential equations and its applications to dynamics was clearly at the center of Poincaré’s mathematical thought; from his first (1878) to his last (1912) paper, he attacked the theory from all possible angles and very seldom let a year pass without publishing a paper on the subject” (DSB XI: 56). “Poincaré’s thesis, which was examined by Bouquet, Bonnet and Darboux, concerned the study of integrals of first-order partial differential equations in the neighbourhood of a singular point. It was his second paper on the theory of differential equations and, as Hadamard remarked, it contained a strong pointer towards his future success with the topic and its applications to celestial mechanics: ‘Even Poincaré’s thesis contained a remarkable result which was destined later to provide him with a powerful tool in his researches in celestial mechanics’” (Barrow-Green, p. 44). “The development of mathematics in the nineteenth century began under the shadow of a giant, Carl Friedrich Gauss; it ended with the domination by a genius of similar magnitude, Henri Poincaré. Both were universal mathematicians in the supreme sense, and both made important contributions to astronomy and mathematical physics … Poincaré remains the most important figure in the theory of differential equations and the mathematician who after Newton did the most remarkable work in celestial mechanics” (DSB XI: 52). Only two copies at auction listed by ABPC/RBH. OCLC lists Harvard and Penn State only in US.

Poincaré was “awarded the degree of docteur des sciences mathématiques from the Sorbonne on 1 August 1879, for a thesis on a delicate problem in the theory of partial differential equations. It was nominally concerned with extending a theorem of Sonya Kovalevskaya’s on partial differential equations in the complex domain to understand the nature of the solutions near a singular point of the equation, along the lines of his study of ordinary differential equations in his [‘Note sur les proprieties des fonctions définies par les equations différentieles,’ *Journal de l’Ecole Polytechnique* 45 (1878), 13-26]. He was successful when the singular points belonged only to particular solutions. In this case he showed that the solutions satisfied algebraic equations whose coefficients were holomorphic functions of the variable [i.e., differentiable in the sense of complex functions]. But when the singularities were intrinsic to the equation he professed himself defeated. Even in the corresponding case for ordinary differential equations there were difficulties that were to occupy him some years later, in his [‘Sur les integrals irrégulières des équations linéaires,’ *Comptes Rendus* 101 (1885), 939-941] and subsequent papers. His thesis also contained important results on lacunary series [those with large gaps between the consecutive powers in the series] and a mild generalization of algebraic functions, which were later to play an important part in his study of complex functions of several variables … Darboux saw a version [of the thesis] in late 1878 and wrote to Poincaré on 4 December: ‘I continue to believe you have written a good thesis, but it seems to me indispensable that you re-edit it completely and correct all the errors of calculation and changes of notation that make it almost unreadable.’ Perhaps something was done, but he later wrote of Poincaré’s thesis that ‘it certainly contained enough results to furnish the material for several good theses. But if one wanted to give a precise idea of the manner in which Poincaré worked, many points in it still need corrections or explanations.’ This Poincaré was never to supply … perhaps because he was already thinking of other novel approaches to problems about differential equations; his thesis was published for the first time in the first volume of his *Oeuvres* (1928) [sic]” (Gray, pp. 159-160).

“Looked at in the context of the theory of differential equations already in existence at the time, Poincaré’s thesis was the natural convergence of two streams of thought. On the one hand, Cauchy, and later Kovalevskaya, had applied Cauchy’s method of majorants to obtain results about the solutions of partial differential equations in the neighbourhood of an ordinary point, while on the other, Briot and Bouquet, and later Fuchs, using similar methods, had studied the solutions of ordinary differential equations in the neighbourhood of a singular point. Poincaré followed both Cauchy, by considering the solutions of partial differential equations, and Briot and Bouquet, by considering these solutions in the neighbourhood of a singular point.

“Poincaré’s analysis divided naturally into two parts according to whether the singularities under consideration were essential. In the case where the singularities were non-essential, he found that the solutions satisfied algebraic equations with coefficients analytic with respect to the variables. Although his treatment concerned a single partial differential equation, his results were analogous to those that would have been obtained by applying the theory to a system of ordinary differential equations, and it was in this analogous form that he applied the results in [‘Sur le problème des trois corps et les équations de la dynamique,’ *Acta Mathematica* 13 (1890), pp. 1-270]” (Barrow-Green, p. 44), his famous work on celestial mechanics.

“With regard to the second and more difficult case concerning the essential singularities, one of the equations he considered was of the form

*X*_{1 }*∂z/∂x*_{1}** + … + ***X*_{n }*∂z/∂x*_{n} = *λ*_{1}

where the *X _{i}*can be expanded as powers of

*x*

_{1}, ...,

*x*, with no constant term, and the first degree terms can be reduced to

_{n}*λ*

_{1}

*x*

_{1}, …,

*λ*

_{n}

*x*

_{n}.He showed that this equation admits an analytic solution in

*x*

_{1}, ... ,

*x*providing, firstly, there is no relation of the form

_{n}*m*

_{2}

*λ*

_{2 }+ … +

*m*

_{n}

*λ*

_{n}=

*λ*

_{1}, where the

*m*are positive integers, and, secondly, that in the plane for the complex variable

_{i}*λ*the convex polygon containing the points

*λ*

_{1}, … ,

*λ*

_{n }does not contain the origin. This latter condition was the result to which Hadamard later referred, and its importance is contained in the fact that it defines a space of non-existence for the solutions of the equations. In [‘Sur le problème des trois corps …’] Poincaré not only used this result explicitly but also further extended it for use in connection with his celebrated asymptotic solutions” (

*ibid*.).

Barrow-Green, *Poincaré and the three-body problem*, Vol. 2, 1997. Boyer & Merzbach, *History of Mathematics*, 1991. Gray, *Henri Poincaré. A Scientific Biography*, 2013.

Large 4to (262 x 210 mm), pp. [iv], 93, [3]. Half-cloth over marbled boards, gilt morocco title label to spine, with original brown printed wrappers bound in (small hole to the front wrapper, probably a paper flaw). A fine copy.

Item #2928

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Price:
$1,500.00
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