46 REPORT—1846. 
dx dy wihy 
+ —_7_ = 0 admits of 
sg , Vi@)" VF) 
an algebraical integral, f (a) being the polynomial a+ 62+ yur +oas pron ) 
x 
Euler (namely that the differential equation 
might be generalised, if we consider the differential equation —7— 
+4 ot he 0. H ks that this i h fons ety 
—=—+ ... + == 0. He remarks that this is perha e on 
VF) Vie) Et SWE” > 
way in which it can be generalised. 
y. Theorems relating to the comparison of different kinds of elliptic fune- 
tions. One of the most remarkable of these is the relation between the 
complete integrals (those, namely, in which the variable a is unity) of the 
first and second kind, the moduli of which are complementary ; that is, the 
sum of the squares of whose moduli is equal to unity. Legendre’s demonstra- 
tion of it is rather indirect, but many others have been since given. Another 
theorem may be mentioned,—that the complete integral of the third kind 
can always be expressed by means of the complete integrals of the first and 
second. A third and most important result shows that in elliptic integrals 
of the third kind we may distinguish two separate species, and that to one 
or other of these any such integral may be reduced. A memorable dis- 
covery of M. Jacobi has greatly increased the importance of this subdivision, 
of which we shall hereafter speak more fully. This part of the subject is, 
a entirely due to Legendre. 
6. The evaluation of elliptic integrals by means of expansions. 
e. The method of successive transformations. The idea of this method 
originated, as we have seen, with Lagrange. It is developed at great length 
by Legendre, with a special reference to the modifications required in apply- 
ing it to the different species of integrals. As Lagrange had shown, the 
series of transformed integrals extending indefinitely both ways conducts us, 
in whichever direction we follow it, towards a transcendent of a lower kind 
than an elliptic integral, or in other words, towards a logarithmic or cireular 
integral. There are thus two modes of approximation, one of which depends 
on aseries of integrals with increasing moduli, and the other on a series 
whose moduli decrease. Thus for the three species of integrals there will 
be in all six approximative processes to be considered. In the case of the 
elliptic integral of the third kind, we have to determine the law of formation 
of the successive parameters 7, 2', &c. 
- ¢, Reductions of transcendents not contained in the general formula 
(« 9: —) to elliptic integrals. 
V1—28 
4. Lastly, applications to various mechanical and geometrical problems. 
This analysis, however slight, will give an idea of the contents of that part 
of the ‘ Exercices de Calcul Intégral’ which relates to elliptic functions, In 
the third volume there are tables for facilitating the calculation of integrals 
of the first and second kind: they are accompanied with an explanation of 
the manner in which they were constructed. The ninth table is one with 
double entry, the two arguments being the angle of the modulus and the 
amplitude. 
13. In 1825 Legendre presented to the Académie des Sciences the first 
volume of his ‘ Traité des Fonctions Elliptiques.’ A great part of this work 
is precisely the same as the ‘ Exercices de Calcul Intégral.’_ By far the most. 
important addition to the theory of elliptic functions consists in the disco- 
very of a new system of successive transformations quite distinct from that 
of Lagrange. 
