ON THE RECENT PROGRESS OF ANALYSIS. 247 



M. Jacob! mentions in a note in Crelle's Journal, that while 

 at Paris he represented, and as he "believed not ineffectually, to 

 Fourier, who was then one of the secretaries of the Institute, that 

 the publication of this memoir would be very acceptable to 

 mathematicians. A long period however was still to elapse 

 before the publication took place. It was possibly retarded by 

 the death of Fourier. In 1841 the memoir appeared in the 

 seventh volume of the Memoires des Savans Etrangers. It was 

 prepared for publication by M. Libri. 



Thus for about fifteen years Abel's general theory remained 

 unpublished ; but in the meanwhile Crelle's Journal was estab- 

 lished, and to the third volume of this he contributed a paper 

 which contains a theorem much less general than the researches 

 he had communicated to the Institute, but far more so than any- 

 thing previously effected in the theory of the comparison of 

 transcendents. This is commonly known as Abel's Theorem. 

 Legendre, in a letter to Abel, speaks thus of the memoir in 

 which it appeared: "Mais le memoire...ayant pour titre Re- 

 marques sur quelques proprietes generates, &c., me parait sur- 

 passer tout ce que vous avez publi6 jusqu'a present par la 

 profondeur de 1'analyse qui y regne ainsi que par la beaute 

 et la ge'ne'ralite' des resultats." In a previous letter, with refer- 

 ence I believe to the same subject, he had remarked, ' Quelle 

 tete que celle d'un jeune Norvegien !' 



Abel's theorem gives a formula for the comparison of all 

 transcendental functions whatever whose differentials are irra- 

 tional from involving the square root of a rational function of x. 



In a very short paper in the fourth volume of Crelle's 

 Journal, which must have been the last written of Abel's pro- 

 ductions, the chief idea of his general theory is stated; and 

 in the second volume of his collected works we find a somewhat 

 fuller development of it, in a paper written before his visit to 

 Paris, but not published during his life-time. 



While Abel's great memoir remained unpublished at Paris, 

 several mathematicians, developing the ideas which he had 

 made known in his contributions to Crelle's Journal, succeeded 

 in establishing results of a greater or less degree of generality. 

 Eesearches of this kind may be presented in a variety of forms, 

 because the algebraical function to be integrated, which we have 

 called y, may be defined or expressed in different ways. For 



