44 



which accurately expresses the series in all its convergent 

 cases, and of which the development gives rise to the series in 

 its general form. When no such function y(ji') really exists, 

 then it is only to the numerical value of an approximate func- 

 tion that our computation tends in particular numerical cases ; 

 as, for instance, in such a case as that considered at p. 39. 



It may be worth while to notice here, as an immediate in- 

 ference from the differential theorem, that when a series, pro- 

 ceeding according to the powers of x, and extending to infi- 

 nity, has its coefficients such that their differences at length 

 become zero, that series is always the development of a rational 

 fraction whose denominator is some integral power of {\±x). 



There is, I think, a mistake committed in always attri- 

 buting this theorem to Euler. It was published by Stirling, 

 in his Methodus Differentialis, so early as 1730 ; and I believe 

 no mention of it occurs in the writings of Euler till long after 

 this date. 



VI. — As far as I know, there is but one other general 

 analytical principle that has been affirmed to give countenance 

 to doctrines opposed to those attempted to be established in 

 the present Paper : the principle, namely, that when an alge- 

 braic expression, for continuous numerical values of the va- 

 riable, approaches continuously to a certain finite numerical 

 value, this value properly expresses the ultimate, or limiting 

 state of that expression. In virtue of this principle, it has 

 been stated* that, " Poisson would admit P- 2H 3^— 4^ -f- 

 . . . = 0, since there is no question that, g being less than 

 unity, the mere arithmetical computer might establish, to any 

 number of decimal places, the identity of 1^—2^^ -|- 3^^^ — . . . 

 and (1— </) (l-|-^)~^."t But I submit that the series here 



* Transactions of the Cambridge Philosophical Society, Part II. 1844. 



f In order that the series P — 2^g -\- S^g^ may become convergent 



after n terms, there must evidently exist the condition 



(^)V < h whence g < (;^)^ 



