REPORT ON CERTAIN BRANCHES OF ANALYSIS. 315 



roots of a reduced equation of (w — 1) dimensions, and employs 

 in the determination of the coefficients of this reduced equation 

 if ~^ fundamental eqaaiions, designated by the Hebrew letter {^, 

 and ?«"~^ others designated by the Greek letter i2. It is un- 

 necessary, however, to enter upon an examination of the truth 

 of processes which the author who proposes them has left un- 

 demonstrated ; and in as much as the application of his method 

 to an equation of 5 dimensions would require the formation of 

 625 fundamental equations of the class Aleph and 125 of the 

 class Omega, and the determination of the greatest common 

 measure of 2 polynomials of 24 and 30 dimensions respectively, 

 it was quite clear that M. Wronski might in perfect safety retire 

 behind an intrenchment of equations and operations of this 

 formidable nature. And this was the position which he took 

 in answer to M. Gergonne, who, in the third volume of the An- 

 nales de Mathematiques, in the modest form of doubts, showed 

 that the form of the roots which he had assumed was not essen- 

 tially different from those which Waring, Bezout, and Euler, 

 had assumed, and which Lagrange had shown to be incompa- 

 tible with the existence of a final reducing equation of the di- 

 mensions assigned to it*. 



The process given by Lagrange for determining the dimen- 

 sions and nature of the final reducing equation has been the 

 touchstone by which all the methods which have been hitherto 

 proposed for the solution of equations have been tried, and will 

 probably continue to serve the same purpose for all similar at- 

 tempts which may be hereafter made. Its illustrious author, 

 however, hesitated to pronounce a decisive opinion respecting 

 the possibility of the problem, contenting himself with demon- 

 strating it to be so, with reference to every method which had 

 been suggested, or which could be shown to arise naturally out 



* The works of Hoene de Wronski were received with extraordinary favour 

 in Portugal, where the Baron Stockier, a mathematician of considerable at- 

 tainments, and other members of the Academy of Sciences became converts 

 to his opinions. There is, in fact, a bold and imposing generality, and appa- 

 rent comprehensiveness of views in his speculations, which are well calculated 

 to deceive a reader whose mind is not fortified by the possession of an extensive 

 and well digested knowledge of analysis. In the year 1817, the Academy of 

 Sciences at Lisbon proposed as a prize, " The demonstration of Wronski 's 

 formulae for the general resolution of equations," which was adjudged in the 

 following year to an excellent refutation of their truth by the academician 

 Evangelista Torriani : it chiefly consists in showing, and that very clearly, 

 that the coefficients of the reducing equation of (« — 1) dimensions, assuming 

 the form of the roots of the equation which Wronski assigned to them, can- 

 not be symmetrical functions of those roots, and therefore cannot be expressed 

 by the coefficients of the primitive equation, whatever be the number, nature 

 and derivation of the fundamental equations }^ and -Q which arc interposed. 



