344' THIRD REPORT 1833. 



^ = e' - 6 a» e«^ 

 ax" 



dx"+^ 



1^ = e^ - 6 «"-^ e-, 

 where ?« is any whole number, or zero. If we now suppose 



and eliminate, by means of this equation, e', we shall get 

 ^L_± = _ 6 (1 _ a) «» e«^ 



and therefore 



</"X rf"+^X 



a quantity which is negative for every real value of x. The 

 conclusion which should be drawn, in conformity with Fourier's 

 principles, is, that all the roots of the equation e'^ — b e"^ — 

 are real, as well as those of its successive derivatives; whilst 

 the fact is, that each of those equations has one real root, and 

 an infinite number of imaginary roots, which are included un- 

 der the formula 



_ log b a" + 2 i Tt V^ — 1 . 

 X — ^ ■ 



1 — a 



In reply to this objection, it has been urged by Fourier that 

 Poisson has not very accurately stated the terms of the propo- 

 sition in question as applicable to such a case *, and also that 

 he has neglected to take into consideration all the roots of the 

 equation. For if we suppose that the substitution of two limits 

 a and b, in a function f (x) and its derivatives, gives results 

 which present the same succession of signs between y("+'') (a;) 

 and/W (^x), then those extreme derivative functions, and those 



• This inaccuracy of statement is rather chargeable upon Fourier himself 

 than upon Poisson, who has certainly failed to notice the necessary limitation 

 of this proposition upon the occasion which gave rise to its application in 

 page 373 of the Thiorie de la Chaleur. 



= b {\ -a) a"+' e"', 



