. 
- 
. ON THE RECENT PROGRESS OF ANALYSIS. 49 
the essential point of M. Jacobi’s reasoning, namely, that the assumed forms 
of U and V satisfy the general condition, laid down at the outset of his de- 
_ monstration, here adverted to. 
In 1828 Legendre published the first supplement to the ‘ Traité des Fone- 
tions Elliptiques,’ &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 explicit than that already noticed. It leaves, I think, no doubt of the 
error into which Legendre had fallen. No notice whatever is taken of the 
_ first part of M. Jacobi’s reasoning: and after remarking that the differential 
equation is satisfied when the double substitution is made, he goes on, “ Tout 
se reduit donc 4 faire cette double substitution dans lintégrale y = = = et 
& examiner si elle est satisfaite.” After showing that it is so, he adds, “ Par 
_ ce procédé trés simple il est constaté que l’équation y = _ satisfait..... 
_ al équation différentielle dont l’intégrale est F (k 9) =p F(h py), ete.” (Trait. 
des Fonct. Ell., iii. p. 10). 
Legendre remarks, that although M. Jacobi’s demonstration rests on “ un 
principe incontestable et trés ingénieux,” 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 condi- 
tions were satisfied, the second would also be so, and then showed that the 
_ forms assigned to U and V satisfied the first condition ; Legendre shows the 
assigned forms are such as to satisfy both conditions, on the connection be- 
_ tween which it is therefore unnecessary for him to dwell. In the third sup- 
; plement to the ‘Traité’des Fonctions Elliptiques,’ Legendre has given an- 
, other 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 demonstration of 
_ this theorem received “le dernier degré de rigueur” *. 
_ __ 14, In 1829 M. Jacobi’s great work on elliptic functions, the ‘ Fundamenta 
Nova Theorie Functionum Ellipticarum,’ was published at Kceenigsberg. It 
contains his researches not merely on the theory of transformation, but also 
_ with respect to other parts of the subject. But the great problem of trans- 
_ formation is the fundamental idea of the whole work; the other parts are 
_ subordinate to it, or at least derived from it. The subject is treated with 
_ great fulness of illustration and in a manner not unlike that of Euler. 
Mz. Jacobi begins by considering the possibility of transforming the ge- 
_ neral transcendent whose differential coefficient is unity divided by the square 
root of a polynomial of the fourth degree. Subsequently, having shown that 
__ this transcendent may be transformed by introducing a new variable y equal 
to the quotient of two integral functions of x, and also that the general 
dy 
tr nscendent may be reduced to one of the form S- Vaapa—eyy 
he proceeds to consider the latter in detail. 
¥ The first step of this reasoning, viz. the possibility of the transformation, 
en on a comparison of the number of the disposable quantities in the 
* Verhulst, Traité Elémentaire des Fonctions Elliptiques. 
1846. E 
