330 THIRD REPORT— 1833. 



there are two real roots equal to a ; and generally, as many of 

 the final functions X, X', X", &c., will disappear, under the same 

 circumstances, as there ai'e roots equal to «. 



4th. If the substitution of a value of a makes one intermediate 

 function X^'"' equal to 0, and one only, and if the result be placed 

 between two signs of the same kind, whether + and + or — and 

 — , then there will be one pair of imaginary roots corresponding to 

 this occurrence ; but if be placed between two unlike signs, + 

 and — or — and +, then there will be no root corresponding to it, 

 unless at the same time X = 0. If the substitution of o makes any 

 number of consecutive derivative functions equal to 0, then, if 

 there be an even number ^p of consecutive zeros, there will be^ 

 or {p — 1) pairs of imaginary roots corresponding, according as 

 they are placed between the same or different signs ; and if there 

 be an odd number 2^^ + 1 of consecutive zeros, then there will 

 be^ + 1 or^j pairs of imaginary roots corresponding, according 

 as they are placed between the same or different signs *. 



The preceding propositions may be easily shown to include 

 the theorem of Des Cartes ; for it is obvious that the substitution 

 of for ar in X and its derivatives will give a succession of signs 

 identical with those of the successive coefficients of X, deficient 

 terms being replaced by 0. If the extreme values a and — ^ 

 be substituted, there will be m permanences in one case and m 

 changes in the second ; it will follow therefore that the number 

 of real and therefore positive roots between « and cannot ex- 

 ceed the number of changes of sign corresponding to a: = 0, or 

 amongst the successive coefficients of the equation ; and that the 

 number of real and therefore negative roots between — /3 and 

 cannot exceed the number of permanences corresponding to 

 X = 0, or of changes between and — /3, which is also identical 

 with the number of successive permanences of sign amongst the 

 coefficients of the equation. 



* I have stated this rule differently from Fourier, whose rule of the double 

 sign appears to me to be superfluous. If we consider the zeros as possessing 

 arbitrary signs, the nature and extent of the ambiguity which they produce 

 will always be determined by the circumstances of their position with respect 

 to the preceding and succeeding sign. 



The rule of the double sign, when one of the derivative functions X', X", X'", 

 &c., becomes equal to zero, is made use of in a memoir by Mr. W. G. Horner, 

 in the Philosophical Transactions for 1819, upon a new method of solving nu- 

 merical equations. This memoir, though very imperfectly developed, and in 

 many parts of it very awkwardly and obscurely expressed, contains many 

 original views, and also a very valuable arithmetical method of extracting the 

 roots of affected equations. It makes also a very near approach to Fourier's 

 method of separating the roots of equations. It is proper to state that 

 Fourier's proposition was known to him as early as 1796 or 1797, as very 

 clearly appears from M. Navier's Preface to his Analyse des Equations Deter- 

 minees, a posthumous work, which appeared in 1831. 



