332 CIRCULAR RELATIVE TO A MATHEMATICAL PRIZE. 



solutely strict manner. Unfortunately we know nothing about this 

 method, except that the starting point for its discovery seems to have 

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

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

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

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

 of earnest and persevering study. 



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

 lem within the period of the competition, the prize might be awarded 

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

 indicated manner and com|)letely 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 suflliciently 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 suflBciently 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. Bnot and Mr. Bouquet have opened the way for such a study 

 by their memoir on this subject (Journal de I'ecole polytechnique, cahier 

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

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

 hausted the difficult 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 division of the 



* See p. 35 of the Panegyric on Lejeune-Diriclilet by Kumnier, Abhaaidlungeu der 

 K. Akademie der Wissenschaften zii Berlin, t860. 



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

 Wissenschaften 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- 

 boux, '2:nie serie, t. IV.) (3) Nachrichten von der K. Gesellschaft der Wissenschaften 

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

 boux, 2mes6rie, t. IV). (4) Borchardt's Journal, Bd.90, p. 71 (also in the Bulletin of 

 Mr. Darboux, 2me serie, t. iv). (5) Abhaudlungen der K. Gesellschaft der Wissenschaf- 

 ten zn Gottiugen, 1881 (Bulletin of Mr. Darbuux, t. V). (6) Sitzungsberichte der K. 

 Akademie der Wissenschaften 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. 



