REPORT ON CERTAIN BRANCHES OF ANALYSIS. 333 



that there is one root of X' = 0, and no root of X = 0, between 

 — 1 and ; that there is one root of X"' = 0, two roots of 

 X" = 0, two roots of X' = 0, and'three roots of X = 0, situated 

 between and 1. It remains to determine whether these three 

 roots are all of them real, or two of them imaginary, and also 

 to assign the limits, in the first case *, between which they are 

 placed. 



In the first place, if imaginary roots exist in the derived, 

 they will exist also in the primitive equation. The converse of 

 this proposition is not necessarily true. 



If the succession of indices be 0, 1, 2, then the succession of 

 signs corresponding to 



X", X', X, orXC' + ^XW Xt-'), 

 will be 



(a) + _ + or — + - 



12 12 



(b) + + + - - - 



There will be one real root between a and b in the equation 

 X' = or X^*"^ = 0, and two roots, whether real or imaginary, 

 corresponding to this interval, in X = or X^*""'^ = 0. 



In the first case, if there be two real roots between a and b, 

 then the curve whose equation 

 is y = X =sf(x), where oa = a, 

 o b =■ b, a n ■= f («), b m =■ 

 f (6), will cut the axis at the 

 points a and /3 between a and 6. 

 The curve will have no point 

 of inflection between a and b, 

 since X" preserves the same sign, whether + or — ; and there 

 will be a point t, where the tangent is parallel to the axis, 

 since X*, in the same interval, changes from + to — , or con- 

 versely, and therefore becomes equal to zero between those 

 limits. In this case, the sum of the subtangents (considered 

 without regard to algebraical signs) will be necessarily less than 

 a b ; and if the interval a bhe subdivided sufficiently, so as to 

 furnish new hmits a' and b', then one or both of these points 

 will sooner or later be found between the points of intersection 

 a and ^, and therefore/ («') and/ (6') will one or both of them 

 change their signs. The analytical expression of those geo- 

 metrical conditions, and therefore of the existence of two real 



f (n\ 



roots, will be, that the sum of the subtangents or quotients f,(\ 



■ * We seek for the limits of the real roots only ; we have no concern with 

 those of the imaginarj' roots or of their moduli. 



