366 
[58 
58. 
NOTES ON THE ABELIAN INTEGRALS.—JACOBI’S SYSTEM OF 
DIFFERENTIAL EQUATIONS. 
[From the Cambridge and Dublin Mathematical Journal, voi. ill. (1848), pp. 51—54.] 
The theory of elliptic functions depends, it is well known, on the differential 
elation 
order), the integral of which was discovered by Euler, though first regularly derived 
from the differential equation by Lagrange. The theory of the Abelian integrals depends 
in like manner, as is proved by Jacobi, in the memoir “ Considerationes generales 
de transcendentibus Abelianis” (Crelle, t. jx. [1832] p. 394) to depend, upon the system 
of equations 
+ 
dy 
*/(№ 
= 0, (fx denoting a rational and integral function of the fourth 
v dx _ „ v xdx ^ x n ~- dx _ 
~ V (A) ’ (A) “ 5 ~ V(A) ~ 
(i). 
where fx is a rational and integral function of the order 2n — 1 or 2n, and the sums 
% contain n terms. 
The integration of this system of equations is of course virtually comprehended in 
Abel’s theorem; the problem was to obtain (n— 1) integrals each of them containing 
a single independent arbitrary constant. One such integral was first obtained by 
Richelot (Crelle, t. xxm. [1842] p. 354), “Ueber die Integration eines merkwürdigen Sys 
tems Differentialgleichungen.” by a method founded on that of Lagrange for the solution 
of Euler’s equation; and a second integral very ingeniously deduced from it. A complete 
system of integrals in the required form is afterwards obtained, not by direct integration, 
but by means of Abel’s theorem : there is this objection to them, however, that any one 
of them contains two roots of the equation fx = 0. The next paper on the subject 
is one by Jacobi, “Demonstratio Nova theorematis Abeliani” (Crelle, t. xxiv. [1842] 
p. 28), in which a complete system of equations is deduced by direct integration,