ON THE RECENT PROGRESS OF ANALYSIS. 67 
formule of transformation of the odd orders founded on elementary proper- 
4 ties of elliptic functions. 
In a historical point of view a notice by M. Jacobi in the eighth volume 
_ of Crelle (p. 413) of the third volume of Legendre’s ‘ Traité des Fonctions 
_ Elliptiques’ is interesting. It was here, I believe, that M. Jacobi first pro- 
_ posed the name of Abelian integrals for the higher transcendents, which we 
_ Shall shortly have occasion to consider. After some account of the contents 
_ of Legendre’s supplements, the first two of which contain the greater part of 
_ M, Jacobi’s earlier researches, he goes on to generalise a remarkable reduc- 
tion given by Legendre at the close of his work. 
21, I turn to one of the very few contributions which English mathema- 
_ ticians have made to the subject of this report, namely, to a paper by Mr. 
_ Ivory, which appeared in the Phil. Trans. for 1831. His design is to give 
_ in asimple form M. Jacobi’s theorem for transformation. The demonstra- 
_ tion is essentially the same as that in the ‘Fundamenta Nova.’ But Mr. 
; Ivory does not set out with assuming y= = U and V being integral fune- 
_ tions of x, but with assuming it equal to the continued product of a number 
_ of elliptic functions (whose arguments are in arithmetical progression), mul- 
_ tiplied by a constant factor. This is one of M. Jacobi’s transcendental ex- 
pressions for y, and the two assumptions are therefore perfectly equivalent 
in the transformations of odd orders; but in those of even orders, or where 
_ the continued product consists of an even number of factors, Mr. Ivory’s 
_ amounts to making y equal to an irrational function of x. Transformations 
__ by irrational substitutions, though long the only kind known (since Lagrange’s 
_ belongs to this class), had not of late been considered in detail. Abel 
_ indeed remarked in the beginning of the general investigation contained in 
a Schumacher’s Journal (No. 138), that the existence of an irrational trans- 
_ formation implied that of a rational one leading to an integral with the same 
modulus as the other. He was, therefore, in seeking for the most general 
modular transformation, exempted from considering irrational substitutions ; 
but in a historical point of view it is interesting to see the connection between 
Lagrange’s transformation and those which have been more recently disco- 
vered*, 
ee: a ser 
Ify ane, FF where 5?-+-c?=1, then 
Mehr tes Ae Sh sna. hub 
Gander 0+) Vas acew) 
‘This is Lagrange’s direct transformation, The corresponding rational transformation is 
_ 1—(1+44) 2? 
Y= Ta (1—b) a” 
~ which satisfies the same differential equation as before. 
Avain 4 dy a ie dr 
=" Vi-~)I-#f) 2 Vi-8)(1—-ex) 
js a 
l+e 
_ (le) 
~ 1lfex?’ 
r2 
