Resolution of Algebraical Equations. 127 



power of x, with its sign changed, will be the required function 

 of the root ; minus the same function of o." 



In Section 3, there is given a method equally simple with 

 the former ones, to obtain the sum of any specified number of 

 the roots; also the sum of any given function of a specified 

 number of the roots. 



In this Section it is also shewn how to find the »«"' least root 

 of an equation, or any function of it, to which are annexed 

 some remarks on that relation of imaginary quantities, which cor- 

 responds to the relation of greater and less in real quantities. 



As the sum of m roots exceeds that of m - 1 by one root, 

 it is clear that by this method we can get all the roots of the 

 equation, as well as any function of any root. 



The principles laid down in the first three Sections are ap- 

 plied in the fourth to the deduction of several theorems of ana- 

 lysis; the theorems of Laplace and Lagrange are simple and 

 almost immediate consequences: a theorem somewhat similar to 

 Lagrange's, which M. Cauchy gives in (Vol. IX. Memoirs of the 

 Institute,) for the sum of any function of all the roots of an 

 equation, I have here shewn holds true for any specified number 

 of the roots as for the whole, though the series only terminates 

 in the latter case, and under particular conditions. 



There is also a theorem given by Burmann, which is of great 

 use in transforming series, but the ordinary demonstration is very 

 long, and may be seen in the notes at the end of Vol. III. 

 Lacroix, Diff. Calc. but in the present method it follows in a 

 few lines. 



When we revert a proposed series, or find the root of an 

 algebraic equation in a series, the law of the latter is not 



