336 THIRD KEPORT — 183^. 



assign their limits, and thus to prepare them for the certain ap- 

 plication of methods of approximation. They constitute a most 

 important element in the theory of numerical equations ; and 

 though they do not enable us to assign the limits of the moduli 

 of the pairs of impossible roots nor to determine their signs of 

 affection, yet they at once indicate both their existence and their 

 number, and thus form the proper pi'eparation, at least for the 

 application of methods, whether tentative or not, for the deter- 

 mination of their values. 



Lagrange, in the fifth chapter of his Resolution des Equa- 

 tions Numeriques, has shown in what manner the equation of 

 the squares of the differences may be apjilied to the deter- 

 mination of these imaginary roots ; and the methods which 

 thence arise are equally complete, in a theoretical sense, with 

 those which are made use of, by the aid of the same equation, 

 for the determination of the limits of the real roots ; and Le- 

 gendre, also, has furnished tentative methods of approximating 

 to then' values. But all such methods are more or less nearly 

 impracticable for equations of high orders ; and the invention 

 of a ready and certain method of separating the imaginary 

 roots of equations, as the basis of processes for approximating 

 to their values, must still be considered as a great desideratum 

 in algebra. 



The method of approximating to the roots of numerical equa- 

 tions, when their limits are assigned, which Lagrange has given, 

 by means of continued fractions, is so well known that it is quite 

 unnecessary to enter upon a detailed examination of its princi- 

 ples. If there is only one real root, included between two con- 

 secutive whole numbers, there will be only one positive root in 

 the several transformed equations, which is greater than 1 , and 

 methods which are certain and sufficiently rapid may be applied 

 to the determination of the several quotients which form the 

 converging fractions. If, however, there are two or more roots 

 included between two consecutive whole numbers, there will be 

 two or more roots of the first transformed equation, and possi- 

 bly, likewise, of the transformed equations which follow which 

 are greater than 1, and which may be placed between two con- 

 secutive whole numbers. The separation of such roots may be 

 effected by the methods of Fourier, which have been explained 

 above ; but when we have once arrived at a transformed equa- 

 tion which has two or more roots greater than 1, no two of which 

 are included between two consecutive whole numbers, then we 

 shall find the same number of sets of successive transformed 

 equations, which will furnish the several sets of quotients to the 

 continued fractions, which represent the roots of the primitive 



