414 On an Improvement in the Analysis of Equations, 



our business would then be to take a portion of only the 

 positive region of this interval, and to employ its limits in the 

 corresponding interval of the original analysis : if two roots 

 were still indicated, within these limits, we may suspect 

 them to be real : if no roots be indicated, we must widen the 

 positive interval in the second analysis : if the roots are real, 

 we shall thus at length inclose them; but if no roots become 

 indicated till we trench upon the negative portion of the in- 

 terval in (B), we may be sure that the roots are imaginary. 



At present I shall give but a single example in illustration. 

 Let the equation be 



^'^ — 4.r3 — 3a? + 23 = 0, 

 the complete analysis of which, by the method of Fourier, is 

 attendecl with a good deal of trouble. (See Analyse des Equa- 

 tions, p. 137.) 



As in this example b is zero, the expression (B) is 



4a;'2 + 3^ — 23; 



and as the only doubtful interval, as ascertained by the partial 

 analysis of Budan (ibid. p. 138), is the interval [1, 10], it is in 

 reference to this alone that we have to examine the signs of 

 the quadratic expression just written. Of this interval, the 

 portion [1,2], being negative, we reject it: taking therefore 

 the interval [3, 10], which is wholly positive, and employing 

 it, instead of the wider interval [1, 10], in Budan's analysis, 

 a root is detected ; and consequently, since no root can lie in 

 the rejected region, the other root must lie in the interval 

 [2, 3] . And thus the character and places of the roots are 

 ascertained. 



It is proper to add, that if the second term be absent from 

 the proposed equation (A), then the rational part of each con- 

 jugate factor will be 



and the expression under the radical will be 



to which all the preceding remarks apply. 



1 need scarcely observe, after what has been shown in my 

 paper in the April Number o!" this Journal, that the conjugate 

 factors here employed form but one pair, out of several that 

 might be used for a similar purpose ; although the forms here 

 given will, as far as they go, be those most eligible; and 

 1 venture to think that, by our availing ourselves of them in 



