PRIZE QUESTIONS. 453 



* by wliole values of A, B, C, D, E, provided that one or more of these values 

 of X really exist, and if not, that the impossibility of their existence be shown. 

 Ejcp/anation. — A method employed by the ctdebrated Pierre de Fermat to 

 reduce to a square A + Bx + C.r^ + Ds^ + Ea;'*, or to a cube the expression, 

 A + Bx + Cx^ + Dx^, is given by P. Jacques de Billy in his work ent.tled Doc- 

 trines analytlcce invcntuin novum, (p. 30 and 31 of tjie edition entitled Diophanti 

 Alcxandrini libri sex, ct de numeris multangulis liber unus, i>)-c. ToJosfs, 

 MDCLXX.) This method is also explained by Leonard Euler in the eighth, 

 ninth, and tenth chapters of the second volume of his work entitled EinJcitung 

 der Algebra, translated into French under the title of Elemcns d'Algebre. 



The XI volume of Memoirs of the Imperial Academy of Sciences of St. Pe- 

 tei'sburg (1830) contains several posthumous memoirs of Euler, relative to the 

 analysis of Diophantus, one of which if) entitled Methodusnova ctfacilis formu- 

 las cubicas ct biquadratitas ad quadratmn reducendi. This method, well consid- 

 ered, is no other, says Jacobi, than that of the muUiplication of elliptic integr-als ; 

 a method already proposed by Euler himself in his Institutions of the integral 

 calculus and elsewhere, in order to resolve algebraically the transcendental equa- 



' , f{x) = a-\-bx+ C3? + dx} -\-ex^. This ob- 



servatiou of Jacobi is found in the XIII volume of the Journal de Mathe- 

 matigues of M. A. L. Crelle (1835) at the article, De usu theoria integralium 

 dlixiticoruvi et integralium Abehanorum in analysi Diophantea. The method 

 given by Fermat for reducing to a square A + Bx + Cx^ + Dx^ + Ea;'' is also stated 

 in the volume entitled Theorie de nombres, 'Sd edition; bij Adricn Marie Le- 

 gendre. Tome ii, Pans, 1830, (p. 123-125.) In a memoir of Lagrange, en- 

 titled Sur quelques j)robl ernes de V analyse de Diopliante, and inserted in the 

 Nouveatix Mejnoires de V Academie royale dvs Sciences et belles-lettres, 1111 ; d 

 Berlin, 1779, a method also is given for resolving into rational numbers the general 

 equations of the third and fourth degree between two indeterminates x, y. 



Nevertheless, these methods are imperfect : 1st, because they already suppose 

 a known solution; 2d, because it is not demonstrated that they furnish all the 

 solutions possible. It is therefore desirable that another should be found in 

 which there should be no need of the knowledge of any solution, and it should 

 be made to appear whether the problem be or be not possible, and, if possible, 

 that all the solutions be given. This would be of great advantage in the theory 

 of numbers, or indetermiuate analysis, and would open the way to important 

 progress, the question having as yet not been satisfied, except in very special 

 cases treated by learned geometers under the above-noticed conditions. It would 

 be conducive also to the progress of other parts of the mathematical sciences, 

 as may readily be seen from the relation, indicated by Jacobi in the memoir 

 already cited, between the problem propounded and the doctrine of elliptical 

 functions. 



Conditions. — Memoirs on the proposed theme should be rendered in Italian, 

 Latin, or French ; no other language is admissible. Each memoir will bear a 

 motto, which shall be repeated on a sealed envelope containing the name and 

 address of the author ; and only that envelope will be opened corresponding to 

 the memoir which shall have obtained the prize. If the authors who receive 

 honorable mention desire that the academy should publish their names, it will 

 be necessary for them to signify their wishes within three months from the day 

 on which the prize is awarded ; at the end of that term the envelopes will bo 

 burned without having been unsealed. The academy has decided that, with the 

 exception of its own thirty members in ordinary, any one, whatever his nation- 

 ality, may compete for the prize. The memoirs and envelopes must be trans- 

 mitted free of postage before the 1st of October, 1866, when the competition will 

 close, and the prize will be awarded in January, 1867. It will consist of a gold 



