332 CIRCULAR RELATIVE TO A MATHEMATICAL PRIZE. 



solutely strict maiiiier. Uufortimately we know uothing about this 

 method, except that the starting i)oiut for its discovery seems to have 

 been the theory of iiifiuitely small oscillations.* It may however be 

 supposed almost with certainty that this method was not based on long 

 and complicated calculations, but on the develojjment of a simple fund- 

 amental idea, which one may reasonably hope to find again by means 

 of earnest and j)ersevering study. 



"However, in case no one should succeed to solve the proposed prob- 

 lem within the period of the competition, the i>rize might be awarded 

 to a work in which some other problem of mechanics is treated in the 

 indicated manner and completely solved. 



"2. Mr. Fuchs has demonstrated in several of his memoirst that 

 there exist uniform functions of two variables which, by their mode of 

 generation, are connected with the ultra elliptical functions, but are 

 more general than these, and which would probably acquire great im- 

 portance for analysis, if their theory were further developed. 



"It is proposed to obtain, in an explicit form, those functions whose 

 existence has been proved by Mr. Fuchs, in a sufficiently general case, 

 so as to allow of an insight into and study of their most essential 

 properties. 



"3. A study of the functions defined by a sufficiently general differ- 

 ential equation of the first order, the first member of which is a rational 

 integral function with respect to the variable, the function, and its first 

 differential coefficient. 



"Mr. Briot and Mr. Bouquet have opened the way for such a study 

 hy their memoir on this subject (Journal de I'^cole polytechnique, cahier 

 o6, pp. 133-198). But the mathematicians acquainted with the results 

 attained by these authors know also that their work has not by far ex- I 

 hausted the diflQcult and important subject which they have first treated. 

 It seems probable that, if fresh inquiries were to be undertaken in the 

 same direction, they might lead to theorems of high interest for analysis. 



" 4. It is well known how much light has been thrown on the general 

 theory of algebraic equations by the study of the special functions to 

 which the division of the circle into equal parts and the diA'ision of the 



* See p. 35 of the Panegyric on Lejeune-Dirichlet by Knmtner, Abhandlungen der 

 K. Akaderaie der Wisseuscliafteii zu Berlin, 1860. 



t These memoirs are to be found in — (1) Nachrichten vou der K. Gesellschaft der 

 ^Vissen8ohaften zu Gottingen, February, 1880, p. 170. (2) Borchardt's Journal, Bd. 

 89, p. 251. (A translation of this memoir is to be found in the Bulletin of Mr. Dar- 

 i)ous,"2:uie serie, t. iv.) (3) Nachrichten von der K. Gesellschaft der Wissenschafteu 

 zu Gottingen, June, 1880, p. 445 (translated into French in the Bulletin of Mr. Dar- 

 boux, 2raes^rie, t. IV). (4) Borchardt's Journal, Bd, 90, p.Tl (also in the Bulletin of 

 Mr. Darbonx, 2me s6rie, t. iv). (5) Abhandlungen der K. Gesellschaft derWissenschuf- 

 ten zu Gottingen, 1881 (Bnlletm of Mr. Darboux, t. v). (6) Sitzungsberichte der K. 

 Akademie der Wissenschafteu zu Berlin, 1883, i, p. 507. (7) The memoir of Mr. Fuchs 

 published in Borchardt's journal, Bd..76, p. 177, has also some bearings on the memoirs 

 quoted. 



I 



