REPORT ON CERTAIN BRANCHES OF ANALYSIS. 305 



essential part of the demonstration requires a double integra- 

 tion between assigned limits, a process against which serious 

 objections may in this instance be raised, independently of its 

 involving analytical truths and principles of too advanced an 

 order. 



The demonstration of Legendre depends upon the possible 

 discovery, by tentative or other means, of values of § and fl, 

 which render P and Q very small ; and subsequently requires 

 us, by the application of the ordinary processes of approxima- 

 tion, to find other values of g and 9, subject to repeated correc- 

 tion, which may render P and Q smaller and smaller, and ulti- 

 mately equal to zero. The objection to this demonstration, if 

 so it may be called, is the absence of any proof of the necessary 

 existence of values of § and fl ; and if they should be shown to 

 exist, it seems to fail in showing that the subsequent correc- 

 tions of their values which this process would assign would really 

 and necessarily increase the required approximations. 



The demonstrations of Cauchy are formed upon the general 

 scheme of that which is given by Legendre, at the same time 

 that they seem to avoid the very serious defects under which 

 that demonstration labours : he shows that (P^ + Q-) must ad- 

 mit of a minimum, and that this minimum value must be zero. 

 The second of the demonstrations differs from the first merely 

 in the manner of establishing the existence and value of this 

 minimum : they both of them appear to me to be quite com- 

 plete and satisfactory. 



It is not very difiicult to establish this fundamental propo- 

 sition by reasonings derived from the geometrical representa- 

 tion of impossible quantities. This was done, though imper- 

 fectly, by M. Argand, in the fifth volume of Gergonne's An^ 

 nales des Mathematiques* , and has been since reconsidered by 

 M. Murey, in a very fanciful work upon the geometrical in- 

 terpretation of imaginary quantities, which was published in 

 1827. It seems to me, however, to be a violation of propriety 

 to make such interpretations which are conventional merely, 

 and not necessary, the foundation of a most important symbo- 

 lical truth, which should be considered as a necessary result of 

 the first principles of algebra, and which ought to admit of de- 

 monstration by the aid of those principles alone. 



General Solution of Equations. — The solution of equations 

 in its most general sense would require the expression of its 

 roots by such functions of their coefficients as were competent 



• In the fourth volume of the same collection there are demonstrations of 

 this fundamental proposition, given by M. Dubourguet and M. Encoutre, 

 which do not appear, however, to merit a more particular notice. 

 1833. X 



