ON THE RECENT PROGRESS OF ANALYSIS. 59 
But elliptic functions are doubly periodic, and therefore the roots of the 
equation in ¢ a are expressible by a formula analogous to the one just written, 
_ but which involves two indeterminate integers corresponding to the two 
_ periodicities of the function, just as p does to the single periodicity 2 7. 
Giving all possible values to these integers, we get »? different values for 
the formula. 
The question now is, how are we to pass from the transcendental repre- 
sentation of these roots to their algebraical expression? Or, in other words, 
how are the relations among the roots deducible from the circumstance of 
their being all included in the same formula, to be made available in effect- 
_ ing the solution of the algebraical equation ? 
The answer to this question is to be found in the following principle: that 
if % wu be such a rational function of w that 
OT AGS A. Aan aan te A 
 @, Y, .z being the roots of an algebraical equation, then any of these quan- 
tities may be expressed in terms of the coefficients of the equation. This 
follows at once from the consideration that we shall have 
1 
X= TAKetxyY tr letate +x 2}> 
& being the number of the roots 2, y,-..2- For the sum within the bracket 
_ being a rational and symmetrical function of the roots, is necessarily expres- 
sible in the coefficients of the equation, and the same is therefore of course 
true of x 2, or of any of the other quantities to which it is equal. 
If, therefore, by means of the relations whichwe know to exist among the 
_ roots of the equation to be solved we can establish the existence of a system 
_ of such functions, x, x’, 9!', &e., each of which retains the same value of 
_ whichever root we suppose it to be a function; and if by combining these 
functions we can ultimately express z in terms of them, the equation is solved, 
since each of these functions may be considered a known quantity. 
_ Such is the general idea of Abel’s method of solution. The principle on 
_ which it depends, namely, the expressibility of any unchangeable function x, 
is one which is frequently met with in investigations similar to that of which 
We are speaking. M. Gauss’s solution of the binomial equation is founded 
upon it. ; 
_ [have already remarked that an important simplification of Abel’s process 
_ was given by M. Jacobi. The result which M. Jacobi has stated without 
_ demonstration may be proved by means of a theorem established by Abel in 
_ the fourth volume of Crelle’s Journal, p. 194. 
_ M. Jacobi shows the existence of a system of n° functions y, x/, &e., by 
(ppowbining which we can immediately express the values of the roots. In the 
_ last of his ‘ Notices’ on elliptic functions we find, as has been said, the ex- 
plicit determination of all the roots. The formula given for this purpose is, 
_ like the former, undemonstrated, and I do not know whether any demonstra- 
_ tion of it has as yet been published; but from a note of M, Liouville, in a 
recent volume of the ‘Comptes Rendus,’ we find that both he and M. Her- 
_ tite have succeeded in proving it. 
_ But in whatever manner the solution is effected it will always involve cer- 
ain transcendental quantities, which are introduced in the expressions of the 
elation subsisting between the different roots. The solution can therefore 
_ be looked on as complete, only if we consider these to be known quantities. 
_ They are the roots of a particular case of the equation to be solved. They 
_ felate to the division of what are called the complete integrals, We may 
_ therefore say that the general case is reduced to this particular one. But 
