348 THIRD REPORT — 1833. 



+ a + + &C., 



X — p X — y 



when a, /3, y, &c., are the roots of the equation, whether real or 

 imaginary, has been shown by Lagrange to be the series fur- 

 nished by this method which is most easily formed, and to be 

 likewise that which converges most rapidly and certainly to a 

 geometrical series in the case of equal roots. In every case the 

 terms of the series of qviotients are alternately greater and less 

 than the root to be determined, and consequently furnish a mea- 

 sure of the accuracy of the approximation. 



This method of approximation is generally less rapid and 

 certain than those which have already been considered, and, 

 as commonly stated, is extremely limited in its application. It 

 is true, as has been shown by Lagrange, that a knowledge of 

 the limits of the roots would enable us to apply it to the deter- 

 mination of all the real roots by means of a series of transformed 

 equations equal to their number, such as is required in the New- 

 tonian method of approximation, and also in that of Lagrange ; 

 but under such circumstances, and with such data, it is more 

 convenient and more expeditious to employ those methods in 

 preference to the one which we are now considering. 



Fourier has shown in what manner this method may be ap- 

 plied to determine all the roots of an equation, whether ima- 

 ginary or real. Let us suppose a, b, c, d, e, &c., to represent 

 the roots of the equation arranged in the order of magnitude, 

 the magnitudes of imaginary roots being estimated by the mag- 

 nitudes of their moduli; and let A, B, C, D, E, &c., be the 

 terms of the recurring series, whose quotients furnish the value 

 of the greatest root,when that root is real. Form, in the se- 

 cond place, a series whose terms are AD — BC, BE — CD, 

 C F — D E, &c., which is also a recurring series, whose quo- 

 tients may be easily shown to approximate to the sum of the 

 two first roots a + b. Again, form a series whose terms are 

 A C - B^ B D - C^, C E - D2, &c., which is also a recurring 

 series, whose successive quotients will approximate to the value 

 of the greatest product a b. In a similar manner, we may de- 

 duce from the primitive recurring series three other recurring 

 series, the terms of the convergent series formed by whose quo- 

 tients will form, in the first series, the sum « + 6 + c of the 

 three first roots ; in the second, the sum of their products two 

 or two, orab + ac + bc; and in the third their continued 

 product a b c : and similarly for four or a greater number of 

 roots. If, therefore, we suppose the first root a to be imaginary, 

 the first series will give no result ; but the values of a + b and 



