375 



If A z = 0, then - 3 A X A Z is a square, in the case of equal roots, and 

 . A x 1 , SA X 



If all the roots of P = 0 are equal, each coefficient of [1] will be zero ; 

 Q 2 - 3PP' being then identically zero. 



Note L 



In the foregoing discussion, we have not specially considered the 

 cases in which equal roots enter an equation, with the exception of 

 what has just been said as to the cubic. Such special consideration of 

 these cases, when actual solution of the equation is the sole object in 

 view, we do not regard as at all necessary ; and we cannot but think 

 that a^ great deal of labour is sometimes expended, with but little profit, 

 in trying to find out, by tedious common measure operations, whether 

 an equation has roots strictly equal or not. 



If equal roots really enter an equation, the approximation to that 

 one of them which always enters singly into an antecedent derived 

 equation, must cause, not only the results in the corresponding column 

 of work to approximate to zero, but also the results in each of the sub- 

 sequent columns, up to the final column, or that which computes 

 f(x). The simultaneous tendency to zero of the results under /(#), 

 fi( x \f*{ x )> & c -» always of course indicates so many roots either 

 accurately equal, or nearly equal ; unless, indeed, the tendency to zero 

 in f(x) should cease, after a certain number of steps, and thus conduct 

 us to the condition of imaginary roots. 



When, however, the approximation to the single real root here al- 

 luded to has been carried on so far that the incomplete development 

 would be regarded as a value sufficiently near to the complete root of 

 f(x) = 0, if this were the only root in the interval ; then, although the 

 approximate root neither separates the other roots of f(x) = 0, nor yet 

 conducts to the condition of imaginary roots, we may, nevertheless* 

 discontinue the development, and may safely regard the value obtained 

 as a close approximation to one, or two, or three, as the case may be, 

 of the values of x which satisfy the equation f(x) = 0. 



For the roots in question, having been developed up to whatever 

 number of figures may have been settled upon at the outset, as sufficient 

 for the purpose in hand, what can it matter whether the superfluous 

 figures which follow those already found to be common to the two or 

 more roots, are the same, or different for those roots? The roots are 

 practically equal if the figures which completely express them are of 

 no practical value beyond those thus far found to coalesce, or to be 



It. T. A. PEOC. VOL. X. 3 E 



