370 



the development exactly as we should do if the doubtful roots were 

 known to be real. Tor in this way we shall invariably arrive at an 

 absolute term in a transformed equation which, if the roots be imagi- 

 nary, will be seen to be irreducible to zero, however far the approxima- 

 tive process be continued ; that is, we shall have evidence that the ab- 

 solute term (the final number) in each successive transformed equation 

 must tend, as the work proceeds, not to zero, but to a finite limit, be- 

 yond which, towards zero, the absolute term cannot pass — a conclusive 

 indication that the roots in the interval we are thus contracting are 

 imaginary. But the tests of imaginarity will generally enable us to 

 resolve the doubt at an earlier stage of the work. 



It is always better, in Horner's process, to develop positive roots 1 

 only ; and, with this object in view, to convert negative roots into posi- 

 tive by changing alternate signs : the passage of a pair of roots 

 will then always be indicated by the loss of two variations, and 

 there can never be a gain of variations. We speak here of the 

 passage of but a single pair of roots in thus continuously proceeding 

 from the inferior towards the superior limit of the interval but, in 

 equations of high degree, several pairs may pass simultaneously, and 

 consequently as many pairs of variations be lost. Such will always 

 happen when there are four, six, &c, equal roots, or when either of the 

 functions is made up of equal quadratic factors, whether the roots of 

 these be real or imaginary. [The consideration of these cases of equal 

 roots is postponed to a Note at the end.] It is scarcely necessary to 

 remark here that the process for computing the function fix), for any 

 value of x, by Horner's method, supplies, in its progress, the computa- 

 tion, in order, of the subordinate functions 



• •••> ~f0), \m, /,(*)...[!] 



for that value of x. If the interval which is doubtful, as respects the 

 equation f(x) = 0, is doubtful also in reference to one or more of the 

 inferior equations f x (x) = 0, \f 2 (x) =. 0, &c. — a circumstance which 

 the rule of signs of Budan will apprise us of upon comparing the 

 signs due to one limit of the interval with those due to the other limit, 

 then the first of the functions [1] — the function f 3 (x) = 0, suppose,* 

 which, when equated to zero, has two doubtful roots in the same inter- 

 val as each of the functions following, up to f(x) = 0, inclusive, must 

 be such that the immediately preceding function f^x) = 0, will have 

 one, and only one, root in this same interval, f And it is always to this 

 real root that our approximation tends as w r e work onwards towards 



^ * For simplicity, we here suppress the fractional multiplier : the roots of an equation 

 being the same whether the significant member of it be multiplied by a number or not, 

 and as it iswith roots only, and not with numerical values of functions, that we are here 

 concerned, the multiplier alluded to may be dismissed, 

 f See " Theory of Equations," p. 170 = 



