258 0^ THE RECENT PROGRESS OF ANALYSIS. 



integrals. Their moduli are complementary ; while the ratio of 

 the A's in the two integrals can be expressed rationally in terms 

 of the sine of the amplitude of one. This circumstance seems 

 to have suggested to Legendre the possibility of generalising 

 the result. He accordingly assumed a relation between the 

 amplitudes of two integrals, of which the equation subsisting in 

 the theorem of which we have been speaking is a particular 

 case ; and showed from hence that a simple relation perfectly 

 similar to that which he had obtained in the particular instance 

 existed between the two integrals, viz. that they bore to each 

 other a ratio independent of their amplitudes. Their moduli 

 are connected by an algebraical equation, but are not comple- 

 mentary. This circumstance therefore now appeared to be un- 

 essential, though in the Exercices the investigation is intro- 

 duced for the sake of exhibiting a case in which an integral 

 may be transformed into another with a complementary mo- 

 dulus. 



Legendre thus obtained a new kind of transformation, which 

 might be repeated any number of times or combined in an in- 

 finite variety of ways with that of Lagrange. To illustrate 

 this he constructed a kind of table a 'damier analytique.' In 

 the central cell is placed the original modulus c. All the moduli 

 contained in the same horizontal row are derivable from one 

 another by Lagrange's scale of moduli; those in each vertical 

 row by the newly discovered scale. He seems to have been 

 very much struck by the infinite variety of transformations of 

 which elliptic integrals admit. The integral of the first kind 

 is especially remarkable, because of the simplicity of the relation 

 which connects it with any of its transformations, viz. that their 

 ratio is independent of the amplitudes. 



Legendre' s second work was, as we have remarked, pre- 

 sented to the Academy in 1825, but it was not published till 

 1827. In the summer of 1827 M. Jacobi announced in Schu- 

 macher's Astronomischen Nachrichten, No. 123, that he was 

 in possession of a general method of transformation for elliptic 

 integrals of the first kind. He was not acquainted with Le- 

 gendre's discovery of a new scale, and as an illustration of the 

 general theorem gave two cases of it, the first being equiva- 

 lent to Legendre's method of transformation. Thus much was 

 announced in a letter to M. Schumacher, dated June 13th ; 



