CIRCULAR RELATIVE TO A MATHEMATICAL PRIZE. 333 



argument of the elliptic functions by a whole number lead up. That 

 remarkable transcendant which is obtained by expressing the module 

 of an elliptic function by the quotient of the periods leads likewise 

 to the modulary equations that have been the origin of entirely new 

 notions and highly important results, as the solution of equations of 

 the fifth degree. But this transcendant is but the first term, a par- 

 ticular case and that the simplest one of an infinite series of new 

 functions iutroduced into science by Mr. Poincar6 under the name of 

 * fonctions fuchsiennes' and successfully applied by him to the integra- 

 tion of liueary differential equations of any order. These functions, 

 which accordingly have a role of manifest importance in analysis, have 

 not as yet been considered from an algebraical point of view as the 

 transcendant of the theory of elliptic functions of which they are the 

 generalization. 



" It is proposed to fill up this gap and to arrive at new equations 

 analogous to the modulary equations by studying, though it were only 

 in a particular case, the formation and properties of the algebraic rela- 

 tions that connect two 'fonctions fuchsiennes' when they have a group 

 in common. 



'' In case none of the memoirs tendered for competition on any of the 

 subjects proposed above should be deemed worthy of the prize, this 

 may be adjudged to a memoir sent in for competition that contains a 

 complete solution of an important question of the theory of functions 

 other than those proposed by the commission. 



"The memoirs offered for competition should be furnished with an 

 epigraph, and, besides, with the author's name and place of residence 

 in a sealed cover, and directed to the chief editor of the Acta Mathe- 

 matica before the first of June, 1888. 



" The memoir to which His Majesty shall be pleased to award the 

 prize as well as that or those memoirs which may be considered by the 

 commission worthy of an honorary mention, will be inserted in the 

 Acta Mathematica, nor can any of them be previously published. 



" The memoirs may be written in any language that the author 

 chooses, but as the members of the commission belong to three differ- 

 ent nations the author ought to subjoin a French translation to his 

 original memoir, in case it is not written in French. If such a trans- 

 lation is not subjoined, the author must allow the commission to have 

 one made for their own use. 



''The Editors of the Acta Mathematica." 



I hope you will take an interest in this communication, and am, sir, 

 yours, very respectfully, 



G. MiTTAGr LEFFLER, 

 Member of the Academy of Science, Professor of the Stockholvi 



University, Chief Editor of the Acta Mathematica. 



