ON THE RECENT PROGRESS OF ANALYSIS. 277 



If, therefore, by means of the relations which we know to 

 exist among the roots of the equation to be solved we can esta- 

 blish the existence of a system of such functions, ^, ^', %", &c., 

 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 x 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 %, 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. 



I 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 

 ^, %', &c., by combining 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 explicit determination of 

 all the roots. The formula given for this purpose is, like the 

 former, undemonstrated, and I do not know whether any demon- 

 stration of it has as yet been published; but from a note of 

 M. Liouville, in a recent volume of the Comptes Hendus, we 

 find that both he and M. Hermite have succeeded in proving it. 



But in whatever manner the solution is effected it will 

 always involve certain transcendental quantities, which are intro- 

 duced in the expressions of the relation 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 relate to the division of what are called the com- 

 plete integrals. We may therefore say that the general case is 

 reduced to this particular one. But the latter is not, except 

 under certain circumstances, soluble, though the solution of the 

 equation on which it depends can be reduced to the solution of 

 certain other equations of lower degrees. 



But for an infinity of particular values of the modulus, the 



