REPORT ON CERTAIN BRANCHES OF ANALYSIS. 307 



the most scrutinizing and laborious examination by nearly all 

 the greatest analysts who have lived in that period, we may be 

 justified in concluding that this failure is rather to be attributed 

 to the essential impossibility of the problem itself than to the 

 want of skill or perseverance on the part of those "-ho have 

 made the attempt. But in the absence of any compete and 

 uncontrovertible proof of this impossibility, the question cannot 

 be considered as concluded, and will still remain open to spe- 

 culations upon the part of those with whom extensive and well- 

 matured knowledge, and a deep conviction founded upon it, have 

 not altogether extinguished hope. 



The different methods which have been proposed for the 

 resolution of cubic and biquadratic equations, and the conse- 

 quences of the extension of their principles to the solution of 

 equations of higher orders, have been subjected to a very de- 

 tailed analysis by Lagrange, in the Berlin Memoirs for 1770 

 and 1771, and in the Notes xiii. and xiv. of his Traits sur la 

 Resolution des Equations Numiriques ; and it would be diffi- 

 cult to refer to any investigations of this great analyst which are 

 better calculated to show the extraordinary power which he 

 possessed of referring methods apparently the most distinct to 

 a common principle of a much higher and more comprehensive 

 generality. In the subsequent remarks which we shall make, 

 we shall rarely have occasion to proceed beyond a notice of the 

 general conclusions to which he has arrived, and to show their 

 bearing upon some later speculations upon the same subject. 



A very slight examination of the principles involved in the 

 solution of the equations of the third and fourth degrees will 

 show them to be inapplicable to those of higher orders. A no- 

 tice of a very few of such methods vdll be quite sufficient for 

 our purpose. 



Thus, the ordinary solution of the cubic equation 

 a^-Sqx + 2r-Q* 

 is made to depend upon that of the following problem : 



" To find two numbers or quantities such that the sum of 

 their cubes shall be equal to 2 r and their product equal to q." 



If we represent the required numbers by u and v, we readily 

 obtain the equation of reduction 



u^ — 2r u^ + ^"^ = 0, 



• This equation may be considered as equally general with 

 a^_Aa;2 + B«— C = 0, 

 in as much as we can pass from one to the other by a very easy transforma- 

 tion ; and the same remark may be extended to equations of higher orders. 

 Such a change of form, however, will determine the applicability or inappli- 

 cability of many of the methods which are proposed for their solution. 



x2 



