ON THE RECENT PROGRESS OF ANALYSIS. 271 



M. Jacobi's mind, but he does not seem to have expressed 

 it. Further on, in the Fundamenta Nova, we find another 

 mode of expressing integrals of the third kind in terms of 

 functions of two elements, but this method also applies only to 

 ' fonctions du troisieme ordre a parametre logarithmique,' the 

 two methods being in fact closely allied. 



Legendre appreciated the importance of this discovery of 

 M. Jacobi. He speaks of it in a letter to Abel, as a ' decou- 

 verte majeure,' but adds that his attempts to extend M. Jacobi's 

 demonstration to the other ^class of integrals of the third kind 

 had been unsuccessful. The same remarks occur in his second 

 supplement (Traite des Fonct. Ell. ill. p. 141). The distinction 

 thus made between the two classes of integrals of the third kind 

 appeared to Legendre sufficient to make it desirable to recognise 

 in all four classes of elliptic integrals, so as to make the division 

 between the two species of the third class coordinate with that 

 between either and the first or second. Legendre says explicitly 

 that M. Jacobi had announced, in making known his discovery, 

 that it applied to functions ' a parametre circulaire.' This how- 

 ever possibly arose from some misconception of M. Jacobi's 

 meaning. Dr Gudermann, in the fourteenth volume of Crelle's 

 Journal, has given it as his opinion that the circular species of 

 integrals of the third kind does not admit of the reduction in 

 question; and remarks, that it occurs much more frequently 

 than the other species in the applications of mathematics to 

 natural philosophy. 



After having discussed at some length, and by new methods, 

 the properties of elliptic integrals of the third kind, M. Jacobi 

 concludes his work by investigating the nature of two new 

 transcendents which present themselves in immediate connexion 

 with the numerator and denominator of the continued product 

 by which sin am u is expressed. One of them however M. 

 Jacobi had already recognised by a distinctive symbol, in con- 

 sequence of its intimate connexion with the theory of integrals 

 of the third kind. 



Such is the outline of this remarkable work : before it ap- 

 peared M. Jacobi gave in the third and fourth volumes of 







(vide ante, p. 255). The specific names are derived from the circumstance that for 

 the former the fundamental formula of addition involves a logarithm, for the 

 latter a circular arc. 



