REPORT ON CERTAIN BRANCHES OF ANALYSIS. 345 



also which are included between them, when considered as 

 equations, will contain the same number of roots, or none, be- 

 tween those limits. This proposition is true, whether the num- 

 ber of derivative functions be finite, as in the case of algebraical 

 equations, or infinite, as in the case of transcendental equations. 

 In the first case, however, it admits of absolute application, in 

 consequence of our arriving at a final derivative, from which 

 the comparison of the signs of the two series of results com- 

 mences. In the second case we can draw no conclusion, in the 

 absence of any difference in the signs of the series of results, 

 in the transition from one derivative function to another, with 

 respect to the number of roots of any of those functions which 

 are included between the given limits: thus, if f{x) = sin a:, 

 we shall have the same series of signs of sin x and of its deri- 

 vatives, however far continued, upon the substitution of the 

 limits a and a + 2 ir, although it is manifest that there are two 

 real roots of sin x = between those limits. The general pro- 

 position, therefore, will, in such a case, authorize us in con- 

 cluding merely that whatever number of roots the equation 

 sin a: = includes between the limits a and a + 2 tt, will be 

 possessed likewise by all its derivative equations between the 

 same limits*. 



There is another point of view, likewise, in which the objec- 

 tion advanced by Poisson may be considered as not altogether 

 applicable to the example which he puts forward. In considering 



the roots of the derivative functions , „ , -^ — ^r-, --: vn , he 



has not included those of the factor e^, which those functions 



1 + — ) = 0, it follows that 



there are an infinite number of equal roots (where a; = — oo ) 

 of e* = 0, which equally reduce three or any number of conse- 

 cutive derivative functions to zero, and to which, therefore, 

 the test of De Gua is no longer applicable. It would follow, 

 therefore, that the existence of imaginary roots in the equation 

 X = is no longer contradictory to Fourier's proposition, even 



* If the transcendental ftinction denoted by / {x) be a determinate function, 

 it will always be possible to assign an interval S, such that the derivative 

 function /« {x) = contains no root, or a determinate number of roots, be- 

 tween a and a + S. If such an interval or succession of intervals can be de- 

 termined for any one derivative function, such as /W (a.), it will become a 

 point of departure for the determination of the number and nature of the roots 

 corresponding to the same interval or intervals for all the other derivative 

 functions which form the superior or inferior terms of the series. In the case 

 of algebraical functions, the point of departure is that derivative function which 

 16 a constant quantity. 



