REPORT ON CERTAIN BRANCHES OF ANALYSIS. 325 



will be less than the least difference of any two of the real roots 

 of the primitive equation, and will consequently furnish us with 

 such a value of A as will enable us to assign their limits. The 

 extreme difficulty, however, of forming the equation of dif- 

 ferences, which becomes nearly impracticable in the case of 

 equations beyond the fourth degree*, renders it nearly, if not 

 altogether, useless for the purposes for which this transforma- 

 tion was intended by the illustrious analysts who first proposed 

 it ; in other words, it is only in a theoretical sense that it can be 

 said to furnish the solution of the problem of determining the 

 limits of the real roots of an equation. 



Cauchy has succeeded in avoiding the necessity of forming 

 the equation of the squares of the differences of the roots, by 

 showing that a value of A may be determined from the last term 

 of this transformed equation, combined with a value of a limit 

 greater than the greatest root of the primitive equation. If we 

 suppose H to represent this term, k to be the superior limit 

 required, and a and b to represent any two roots of the equa- 

 tion, whether real or imaginary, then he has shown that their 

 difference a — b, or the modulus of their difference, will be 



will be a superior limit to the roots. An inferior limit (without reference to 

 algebraical sign) may be readily found by the same process by the formation 

 of the equation whose roots are the reciprocals of the former. 



M. Bret, in the sixth volume of Gergonne's Avnales des Mathematiques, has 

 investigated other superior limits of the roots of equations, which admit of 

 very easy application, and which likewise give results which are generally not 

 very remote from the truth. One of these limits is furnished by the following 

 theorem : " If we add to vnity a series of fractions whose numerators are the 

 successive negative coefficients, taken positively, and whose denominators 

 are the sums of the positive coefficients, including that of the first term, the 

 greatest of the resulting values will be a superior limit of the roots of the 

 equation." Thus, in the equation 



2 x^ + 11 x^ — 10 aS _ 26 a;-! + 31 a.3 + 72 a;2 — 230 x — 348 = 0, 



the number 4, which is equal to the greatest of the quantities 



1+2^,1 + ^.1 + ^,1 + ^. 



13 13 11(3' 116 



is a superior limit required ; and if we change the signs of the alternate terms, 



we shall have 1 + -— , or 7, a superior limit of the roots of the resulting 



equation : it will follow, therefore, that all the real roots of the first equation 

 will be included between 4 and — 7- Other methods are proposed in the 

 same memoir which are not equally new or equally simple with the one just 

 given, and which I do not think it necessary to notice. 



* Waring, as is well known, gave the transformed equation of the 10th de- 

 gree, whose roots were the squares of the differences of the roots of a general 

 equation of the fifth degree, wanting its second term : it involves 94 different 

 combinations of the coefficients of the original equation, many of them with 

 large numerical coefficients. 



