222 ON THE INSUFFICIENCY OF TAYLOR's THEOREM, 



important properties, and which enter as coefficients of the development 



P + Ph^ + Ph' + Ph^ . . Ph^ . + R, 

 that possesses the property, already mentioned, of permitting one term to be 

 rendered greater than all the remainder. 



To prove the generality of such an expansion became, therefore, a problem 

 of moment, and as, in attempting to solve it, Lagrange had fallen into a ma- 

 terial error, Poisson resumed the subject in a memoir, the substance of which 

 may be found in the large work of Lacroix. 



It was soon felt, however, that Poisson's demonstration proved too much; 

 it was seen that his results, if demonstrated at all, should be true of every spe- 

 cies of function, and that deviations from truths so established would not be 

 ' exceptions, but anomalies. The force of these objections was increased by a 

 discovery, partly his own, that, in a similar case, the expansion for sin n x, 

 mathematicians, by adopting such loose reasoning, had fallen into considerable 

 mistakes. And these doubts were again re-enforced when it was found that, in 

 that celebrated development used by Laplace to investigate the theory of the 

 planets, as well as in the case of the numerous periodic series that began now 

 to be employed, these vague methods of development had introduced uncer- 

 tainty into the most profound inquiries, and thrown doubt upon results of the 

 highest importance. 



Hence arose both Cauchy's laborious reinvestigation of the higher analysis, 

 and those numerous attempts which we have recently seen made, to demonstrate, 

 in a more exact manner, the fundamental expansions of the science. 



Among these it must be acknowledged that one of the most elegant and 

 happy is Poisson's investigation, gi^^en in his Traite de Chaleur, of Fou- 

 rier's Theorem for periodic developments. And yet, even here, it must be 

 allowed that a demonstration is still wanting that shall, at the same time, be 

 more simple, and connect the result more readily with the elementary arrange- 

 ments of numerical logic. 



That such is the case with regard to the theorem that forms the subject of 

 this paper is, I have no doubt, generally felt; and to that theorem I now 

 return. 



Lagrange, after an unsuccessful attempt to prove that fractional powers do 

 not enter the development, commences his demonstration by assuming an ex- 

 pansion of the form 



F(x + h) = F (x)+h.P; 



