304 THIRD REPORT — 1833. 



that in every instance, when roots of equations cease to be real, ' 

 they will assume the form m + ?? V^— 1. 



This demonstration is not merely indirect, but it does not 

 arise naturally from the question to be investigated. It seems 

 likewise to assume the existence of some algebraical form which 

 expresses the value of the root in terms of the coefficients of 

 the equation, an assumption which, as will afterwards be seen, 

 it M'ould be difficult to justify by any a priori considerations. 

 The illustrious author hmiself seems to have felt the full force 

 of these objections, and he proceeds therefore in the following 

 Note to prove that every polynomial of a rational form will ad- 

 mit of rational divisors of the first or second degree. The de- 

 monstration which he has given is founded upon the theory of 

 symmetrical functions, and shows that the coefficients of such 

 a divisor may be made to depend severally upon equations all 

 whose coefficients are rational functions of the coefficients of 

 the polynomial dividend. Whatever be the degree of parity of 

 the number which expresses the dimensions of this polynome, 

 he shows the possibility of the coefficients of this quadratic di- 

 visor, which is the capital conclusion in the theory. It ought 

 to be observed, however, that the whole theory of the compo- 

 sition of equations is so much involved in the different steps of 

 this investigation, or, at all events, that so little provision is 

 made in conducting it to guard against the assumption of 

 this truth, that we should not be justified in considering this 

 demonstration as perfectly independent or as furnishing an 

 adequate foundation for so important a conclusion. If we view 

 it, however, simply with reference to the problem for exhibiting 

 the nature of the law of dependence which connects the coeffi- 

 cients of the polynomial factor with those of the original poly- 

 nomial dividend, it must still be considered as an investigation 

 of no inconsiderable importance, as bearing upon the general 

 theory of the solution and depression of equations. 



The second of the proofs given by Gauss, the proof of Le- 

 gendre, and both of those which have been given by Cauchy, 

 belong to the second class of demonstrations to which we 

 have referred above. Assuming the root to be represented 

 by p (cos 6 + 'S/ — 1 sin 9), the equation is reduced to the form 

 P + Q ^/^-i, or ^(P2 + Q2) . (cos <p + V^l sin (f); and the 

 object of the demonstration is to show that there exist neces- 

 sarily real values of p and 9, which make P^ + Q'^ = 0. This 

 is effected by Gauss by processes which are somewhat syn- 

 thetical in their form, and such as do not arise very natu- 

 rally or directly from the problem to be investigated ; and the 



