372 



c insri-'irive rin::b::is ~1~ in which dc\ib: enters ; and let us assume 

 that the two roots off m (x) = 0, in the interval under examination, are 

 imaginary : then the operation of continuous development, as explained 

 above, will conduct us eventually to a transformed equation, such that 

 the three coefficients under the functions, 



will satisfy the condition of imaginarity ; that is, the triad will indicate 

 the presence of a primary pair of imaginary roots in the transformed 

 equation, as may he proved as follows : 



The approximation being to the real root (r) of f m ^(x) = 0, lying in 

 the contracting interval, the coefficient under f m ^{x) will continue 

 tending to zero, whilst the coefficient under f m (x) is approaching a 

 finite limit. Moreover, in the passage over the interval [r-c, r + £], 

 c may become so small that, in the terms of the transformed equation, 

 up to the term under f m+2 (z) inclusive, no variation shall be lost: but, 

 taking in the two terms next following, two variations are lost in the 

 passage over the root r, since f m ^{x) changes sign in this passage, 

 whilst the signs of the preceding functions, as well as the sign of f m {x), 

 remain unchanged. Xow this cannot possibly be unless for the trans- 

 formation r — t the collocation of signs under the three functions is either 

 •r - +, or - + - ; for otherwise there could not be two variations to 

 lose. Hence, in passing over the interval, fr - c, r + o], the signs of 

 the first and third of the coefficients under the above three functions 

 continue to be like signs; and as the middle coefficient vanishes in 

 this interval, it follows that not only at, but before and after this 

 evanescence, the square of the middle one of the three coefficients must 

 be less than the product of the other two coefficients. The triad must 

 therefore satisfy the condition of imaginarity ; and must do so all the 

 earlier in the process of continuous development, inasmuch as the 

 square has for multiplier a number less than the multiplier for the pro- 

 duct by the number n + 1 (formulas 5, p. 344). So soon as the triad of 

 coefficients satisfies this condition of inequality, a stop may be put to 

 the work, provided but one pair of doubtful roots lies in the interval. 

 "We should know that, in the remaining portion of the interval, a pair 

 of imaginary roots would exist for each succeeding function. If, how- 

 ever, there are other pairs of roots in the interval under examina- 

 tion, the transformation must be completed, and the development be 

 proceeded with, in order to ascertain, from the variations lost between 

 r - c and r + B, whether additional pairs of imaginary roots are also in- 

 dicated by the passage of the root r. 



Whatever pairs, besides these additional pairs (if any), may still be 

 indicated in the original interval \_a, J], they are to be sought in the 

 partial interval [r, 5], by proceeding in the same way as at first. We 

 have only further to observe that, when the leading triad of the proposed 

 equation satisfies the condition at page 343, we know that the leading 

 triad for every transformation will also satisfy it (27). But the pair of ima- 

 ginary roots in f (x) = (3, which this triad indicates, lies in the interval 



