324 THIRD REPORT — 1833. 



tion depressed by a number of units equal to the number of 

 such factors. We might suppose, therefore, in all cases, that 

 the roots of the equation to be solved were unequal to each 

 other ; but if it should not be considered necessary to perform 

 the previous operations which are required for the detection 

 and separation of the equal roots, the failure of the methods of 

 approximation or other peculiar circumstances connected with 

 the determination of the limits of the roots, would indicate their 

 existence, and at once direct us to the specific operations upon 

 which their determination depends. 



If we suppose, therefore, the equal roots to be thus separated 

 from the equation to be solved, and if we assume a quantity 

 A which is less than the least difference of the unequal roots, 

 then the substitution of the terms of the series 

 k A,(]i — \) A,. . , . 2 A, A, 0, - A, -2 A, ... . - k^ A, 



where ^ J is greater than the greatest root, and — k^ A less than 

 the least root*, will give a series of results, amongst which the 

 number of changes of sign from + to — and from — to + will 

 be as many as the number of real roots, and no more ; and v/here 

 the pairs of consecutive terms of the series of multiples of A 

 which correspond to each change of sign are limits to the seve- 

 ral real roots of the equation. This is the principle of the me- 

 thod of determining the limits of the real roots which was first 

 proposed by Waring, and which has been brought into practical 

 operation by Lagrange and Cauchy. It remains to explain the 

 different methods which have been proposed for the purpose of 

 determining the value of A. 



Waring first, and subsequently Lagrange, proposed for this 

 purpose the formation of the equation whose roots are the 

 squares of the differences of the roots of the given equation. If 

 we subsequently transform this equation into one whose roots 

 are the reciprocals of its roots, and determine a limit / greater 



than the greatest root of this transformed equation f, then— -r^ 



* A negative root is always considered as less than a positive root, unless 

 the consideration of the signs of affection is expressly excluded. 



f Newton proposed for this purpose the formation of the equation ■whose 

 roots are x — e, and where e is determined by trial of such a magnitude that 

 all the coefficients of the equation may become positive. In such a case e is the 

 limit required. Maclaurin proved that the same property would belong to the 

 greatest negative coefficient of the equation increased by 1. Cauchy, in his 

 Cours d' Analyse, Note iii., and in his Exercices des Mathematiques, has shown 

 that if the coefficients of the equation, without reference to their sign, be 

 Ai Aj, . . Am, and if n be the number of such coefficients which are different 

 from zero, then that the greatest of the quantities 





