371 



f(x). It is thus that the roots of fix) = 0, lying in the same interval, 

 are separated, when they are real, and when not real, are shown to be 

 imaginary by the continuous tendency of f(x), after a certain stage of 

 the process, not to zero, but to a finite limit ; and it is obvious that such 

 tendency there must necessarily be whenever the roots of fix) = 0, in the 

 interval under examination are imaginary, whether the roots of f z (x) = 0, 

 in that interval be imaginary or not. If these latter roots be real, the 

 process will separate them ; if imaginary, f 3 (x) will itself also tend to a 

 finite limit ; and a pair of imaginaries inf(x) = 0 will be indicated. In 

 the former case, after the passage of the real root of f z (x) = 0, if a real root 

 of f(x) = 0 have not also passed, the process is to be renewed, the ap- 

 proximation being now directed to the development of the remaining 

 single root of f 3 (%) = 0 in the remaining interval ; just as at first it was 

 directed to the development of the single root, in the original interval, 

 of fi(x) - 0 ; until, in the case of the roots of f 2 (x) = 0 continuing 

 doubtful, notwithstanding this further contraction of the interval, these, 

 if real, become separated ; and so on, up to f x {x) = 0, and f (x) = 0. 



In this way, the doubtful roots, if they turn out to be real, are con- 

 tinuously approximated to, however closely they may lie together ; and 

 we now proceed to show that the criteria established at the outset of 

 this paper — without even regarding the tendency of the absolute num- 

 ber,* — can never fail to detect their existence whenever the doubtful 

 roots are imaginary : — to show, in fact, that whatever be the character 

 of a pair of imaginary roots of the proposed equation, that pair will al- 

 ways be replaced by, or give rise to, a primary pair in a more or less 

 remote transformed equation; and this, we think, is an important 

 truth. 



In order to prove it, however, we must premise, what has been clearly 

 enough proved by Fourier, that in the operation of continuous develop- 

 ment, briefly described above, the limits of the doubtful roots become so 

 contracted as we proceed, that not only do those limits exclude all 

 roots except one root, of the function (taking the series of functions 

 from right to left) immediately beyond the last of [1], into which the 

 doubt enters, but they also exclude every root of the immediately next 

 following function : in other words, the interval becomes at length so 

 contracted that in the passage over it, while two variations are lost in 

 the series of signs under the functions [1], reckoning onwards from left 

 to right, up to the above-mentioned doubtful function inclusive, only one 

 variation is lost in the series terminating at the immediately antecedent 

 function : and no variation at all is lost in the series ending at the 

 function immediately before this.f 



(31). Now let f m (x) be the function described above as the last of the 



* Nevertheless, it is always advisable to take note of this tendency as the approxi- 

 mation proceeds. The fluctuations of the absolute number, in its passage from one 

 transformation to another, may give early indication of the true character of the doubt- 

 ful roots, although it be not to a root of/i(x) = 0 that the approximation is directed, 



f See " Theory of Equations," p. 172, et seq. 



