REPORT ON CERTAIN BRANCHES OF ANALYSIS. 349 



of a h, which are given by the two first recurring series de- 

 rived from the primitive recurring series, will enable us to de- 

 termine their separate values : in both cases the series of quo- 

 tients is convergent. 



If the third root be real, the third series of derived quotients 

 is convergent ; if not, the fourth series will be so, and so on as 

 far as we wish to proceed. 



These propositions have been merely announced by Fourier 

 in his Introduction. The chapter of his work, which contains 

 the demonstrations, has not yet been published. 



If the root of an equation be determined approximately, the 

 equation may be depressed, and the general processes of solu- 

 tion or of approximation may be applied to find the roots of 

 the quotient of the division. Thus, in the equation 

 a;3 _ 3 ^ + 2-0000001 = 0, 



one of the roots is very nearly equal to 1, if we divide the 

 equation by a: — 1, and neglect the small remainder which re- 

 sults from the division, we shall get the quotient 



a;^ - ar - 2 = (a; - 1) (a: + 2) = 0, 



whose roots are 1 and — 2 ; or we may suppose one of the 

 roots to be 1*0001, the second "9999, and the third —2; or 

 we may suppose two of the roots to be imaginary, namely, 

 1 + '0001 V — \. All these roots are approximate values of 

 the roots of the equation, which different processes, whether 

 tentative or direct, may determine : and it is obvious that when 

 two roots are equal, or nearly so, an inaccuracy of the approxi- 

 mation to those roots which are employed in the depression of 

 the primitive equation may convert real roots into imaginary, 

 or conversely. Such consequences will never follow when the 

 limits and nature of the roots are previously ascertained, and 

 every root is determined independently of the rest ; but it is 

 not very easy to prevent their occurrence when methods of ap- 

 proximation are applied without any previous inquiries into the 

 nature and limits of the roots, though the resulting conversion 

 of imaginary roots into real, and of real roots into imaginary, 

 may not deprive them of the character of true approximations 

 to the values of the roots which are required to be determined. 

 If the limits of the roots of an equation F ar = be assigned, 

 and if the Newtonian method of approximation be applied con- 

 tinually to one of these limits a, we should obtain, for the value 

 of the root, the series* 



a - «'F« + i^(F«r - n^(F«)' + &«•' 

 ♦ Lagrange, Resolution des Equations Numeriques, Note xi. 



