ECONOMY OF LABOUR IN MATHEMATICS 211 



divide by x — 1 and neglect the small remainder, we have 

 the equation 



x* -f x — 2 = o 



whose roots are 1 and — 2 ; or we may suppose one of the 

 roots to be roooi, the second '9999, and the third — 2 ; or 

 we may suppose two roots to be imaginary, namely 

 1 ± • 000 In/— 1. 



It is thus obvious that if two or more roots are nearly 

 equal, an inaccuracy of the approximation to those roots 

 which are employed in the depression of the primitive equation 

 may convert real roots into imaginary, or conversely. 



De Morgan in 1855 gives the following problem : " Given 

 an equation with a parcel of equal roots, and given certain 

 infinitesimal alterations in the coefficients, required to know 

 what becomes of the equal roots ; how many remain equal, 

 and how many are (a ± &</— i)ed to invent a new participle." 



A consideration of this question by Kronecker (1890- 

 1891) has led him to the following theorem. 



Every rational integral function f (x) of the nth degree with 

 rational numerical coefficients can, by a variation of the co- 

 efficients which is definite in character, but as small as one may 

 please, be reduced to a product of v linear and (n — v)/2 quad- 

 ratic factors, all with rational coefficients, v being an integer 

 determined by the coefficients of f(x). 



Besides this difficulty of separating the roots of an equation 

 when the roots are nearly equal in value, algebraists have 

 occupied themselves with the search for rules, by means of 

 which the limits between which the roots lie can be obtained, 

 and many interesting properties of the equations have been 

 found. Those who are interested in the subject should consult 

 a paper by M. Bret in the sixth volume of Gergonne's Annates 

 des Mathematiques where several methods of finding the limits 

 of roots are given. One of the most interesting is the following : 



" If we add to unity 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 result- 

 ing values will be a superior limit of the roots of the 

 equation." 



