ON THE RECENT PROGRESS OF ANALYSIS. 47 
In the earlier work Legendre had shown that a certain transcendent might 
be expressed in two ways by means of elliptic integrals of the first kind. 
Comparing the two results, he obtained a very simple relation between the 
two elliptic 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 Le- 
= 
_ kind) F(ka)=MF (ay), provided that y and a vanish together. The 
oe : ; : iad Sues 
Pilear that by means of a solution of it we transform the elliptic integral 
‘es 
iv 
gendre the possibility of generalising the result. He accordingly assumed a 
relation between the amplitudes of two integrals, of which the equation sub- 
sisting 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 unessential, 
though in the ‘ Exercices’ the investigation is introduced for the sake of ex- 
hibiting a case in which an integral may be transformed into another with a 
complementary modulus. 
Legendre thus obtained a new kind of transformation, which might be re« 
peated any number of times or combined in an infinite variety of ways with 
that of Lagrange. To illustrate this he constructed a kind of table—a “ da- 
mier 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 rela- 
tion 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, presented to the Aca- 
demy in 1825, but it was not published till 1827. In the summer of 1827 
_ M. Jacobi announced in Schumacher’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 Legendre’s 
discovery of a new scale, and as an illustration of the general theorem gave 
two cases of it, the first being equivalent to Legendre’s method of transfor- 
mation. Thus much was announced in a letter to M. Schumacher, dated 
June 13th; but in one of a later date (August 2nd) he gave a formal 
enunciation of his theorem, but without demonstration. The two commu- 
nications appear consecutively (Ast. Nach. vi. p. 33). 
In No. 127 of the Nachrichten, vi. p. 133, M. Jacobi gave a demonstra- 
- tion of his theorein. 
If we Gan so determine y in the terms of « as to satisfy the differential 
equation 
oo ey op Gg 
Vv (U—y) i—aty)  M V¥(—a*) Ish a) 
“it is manifest that we shall have (F denoting the elliptic integral of the first 
(M being constant), 
question therefore is, how may the differential equation be satisfied, for it is 
F(& 2) into another, viz. into F (Ay). 
__M. Jacobi shows that if y be equal to a U and V being integral funes 
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