The Three-Body Problem by Catherine Shaw Read Free Book Online

geometers familiar with the results discovered by these authors also know that their work is far from having
exhausted the difficult and important subject which they were the first to investigate. It appears probable that new research undertaken in the same direction could lead to propositions of great interest to analysis
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    4. We know what light was shed on the general theory of algebraic equations by the study of those special equations arising from the division of the circle into equal parts, and the division by an integer of the argument of the elliptic functions. The remarkable transcendental number obtained by expressing the module of the theory of elliptic functions by the quotient of the periods similarly leads to the modular equations, which have been the source of absolutely new notions, and to results of great importance such as the solution of the fifth degree equation. But this transcendental number is just the first term, the simplest special case of an infinite series of new functions which M. POINCARÉ has introduced to science under the name of Fuchsian functions, and applied with success to the integration of linear differential equations of arbitrary order. These functions, which play a role of manifest importance in Analysis, have not yet been considered from the point of view of algebra, as the transcendental associated to the theory of elliptic functions, of which they are the generalisation. We propose to fill this lacuna and to obtain new equations analogous to modular equations, by studying, even in just a
special case, the formation and the properties of the algebraic relations relating two Fuchsian functions when they have a common group.
    ‘For those in the know,’ Mr Morrison told us, ‘this Henri Poincaré is considered sure to win the competition. He is a kind of genius, and all of his work is exactly round about the questions proposed above, all four of them, really; he might try his hand at whichever he pleases. He is much admired in Sweden, look – he has published two articles in this very volume. He was formerly a student of Hermite, the commissioner from Paris.’ And he turned some pages, and showed me the first mathematical article in the volume, written in French by this very H. Poincaré, and whose title was precisely ‘On a theorem of Mr Fuchs’, before concluding his translation of the announcement.
    In the case where none of the memoirs presented for the competition on one of the proposed subjects would be found worthy of the prize, this can be attributed to a memoir submitted to the competition, which contains a complete solution of an important question of the theory of functions, which is not one of those proposed by the commission.
    The memoirs submitted to the competition must be equipped with an epigraph and with the name and address of the author in a sealed envelope addressed to the Chief Editor of
Acta Mathematica
before the 1st of June 1888.
    The memoir for which HIS MAJESTY will deign to attribute the prize, as well as that or those memoirs which the commission will consider worthy of an honourable mention, will be inserted into
Acta Mathematica,
and none of them must be published beforehand.
    The memoirs may be redacted in whatever language the author wishes, but as the members of the commission belong to three different countries, the author must provide a French translation together with his original manuscript if the memoir is not already written in French. If no such translation is included, the author must accept that the commission has one made for its own use.
    The Chief Editor
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    ‘The 1st of June, why that is in just three months!’ remarked Emily. ‘Are you submitting a memoir to the competition, Uncle Charles?’
    ‘I? Why no, absolutely not,’ he exclaimed. ‘I know next to nothing about the questions posed here. There are not many people in England nowadays who would be capable of seriously solving them, although if there are any at all, they