308 THIRD REPORT — 1833. 



certain real conditions : those conditions ai-e found to be iden- 

 tical with those which the unknown quantity, or, in other words, 

 the root in an equation of n dimensions, is required to satisfy. 

 The object of the proof above given is to show that it is always 

 possible to find n real magnitudes with known affections which 

 are competent to satisfy these conditions ; and those quantities, 

 therefore, are of such a kind that the equation, whose roots 

 they are, is always resolvible into real quadratic factors ; a most 

 important conclusion, which the greatest analysts have laboured 

 to deduce by methods which have not been, in most cases at 

 least, free from very serious objections. 



There are two classes of demonstrations which have been 

 given of this fundamental proposition in the theory of equations. 

 The first class comprehends those in which the form of the 

 roots is determined from the conditions which they are required 

 to satisfy ; the second class, those in which the form of the 

 roots is assumed to be comprehended under different values of 

 p and fl in the expression p (cos 9 + V^ — 1 sin 6), and it is shown 

 that they are competent to satisfy the conditions of the equa- 

 tion. To the first class belongs the demonstration given above ; 

 those given by Lagrange in notes ix. and x. to his Resolution des 

 Equations Num4riques ; the first of those given by Gauss in the 

 Gottingen Transactiotis for 1816*; and by Mr. Ivory in his 

 article on Equations in the Stipplement to the Kncyclopcedia 

 Britannica. To the second class belongs the second demon- 

 stration given by Gauss in the same volume of the Gottingen 

 Transactions; by Legendre in the 14th section of the first 

 Part of his Theorie des Nombres ; by Cauchy in the 18th 

 cahier of the Journal de VEcole Polytechnique ; and subse- 

 quently under a slightly different form in his Cours d' Analyse 

 Alg4brique. 



The first of the demonstrations given by Gauss, like many 

 other writings of that great analyst, is extremely difficult to 

 follow, in consequence of the want of distinct enunciations of 

 the propositions to be proved, and still more from their not 

 always succeeding each other in the natural order of investi- 

 gation. It requires the aid likewise of principles, or rather of 

 processes, which are too far advanced in the order of the re- 

 sults of algebra to be properly employed in the establishment 

 of a proposition which is elementary in the order of truths, 

 though it may not be so in the order of difficulty. If we may 



• There is another demonstration by Gauss, published in 1799, which I 

 have never seen. In his Preface to his Demonsiratio Nova Altera he speaks 

 of its being founded partly on geometrical considerations, and in other re- 

 spects as involving very different principles from the second. 



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