ON THE RECENT PROGRESS OF ANALYSIS. 261 



namely, that the assumed forms of U and V satisfy the general 

 condition, laid down at the outset of his demonstration, here 

 adverted to. 



In 1828, Legendre published the first supplement to the 

 Traite des Fonctions ElUptiques, &c. It contains an account 

 of the researches of M. Jacobi, and of a memoir by Abel 

 inserted in the third volume of Crelle's Journal. The account 

 here given of M. Jacobi' s demonstration is fuller and more ex- 

 plicit than that already noticed. It leaves, I think, no doubt 

 of the error into which Leggndre had fallen. No notice what- 

 ever is taken of the first part of M. Jacobi's reasoning: arid 

 after remarking that the differential equation is satisfied when 

 the double substitution is made, he goes on, ' Tout se reduit 



done a faire cette double substitution dans I'inte'grale y --y 



et a examiner si elle est satisfaite.' After showing that it is so, 

 lie adds, ' Par ce proce'de tres simple il est constate que I'e'qua- 



x U 



tion y - -y. satis fait ... a 1'dquation differentielle dont 1' in- 

 tegrate est F (&</>) = pF (h-p), etc.' (Trait, des Fonct. Ell. ill. 

 p. 10.) 



Legendre remarks, that although M. Jacobi's demonstration 

 rests on ' un principe incontestable et tres inge'nieux,' it is still 

 desirable to have another verification of so important a theorem. 

 He accordingly gives an original demonstration of it, which is 

 however more nearly allied to M. Jacobi's than to him it seemed 

 to be. This demonstration had already been hinted at in his 

 communication to the Nachrichten. The principal difference is, 

 that while M. Jacobi proved generally that if the first of the 

 two required conditions were satisfied, the second would also be 

 so, and then showed that the forms assigned to U and V satis- 

 fied the first condition ; Legendre shows the assigned forms are 

 such as to satisfy both conditions, on the connection between 

 which it is therefore unnecessary for him to dwell. In the 

 third supplement to the Traite des Fonctions Elliptiques, Le- 

 gendre has given another demonstration of M. Jacobi's theorem, 

 remarking that it is both more rigorous and more like M. 

 Jacobi's than that which he had first given. I have thought 

 it necessary to make these remarks, because it has been said 

 that it was in the supplements to Legendre's work that the 



