REPORT ON CERTAIN BRANCHES OF ANALYSIS. 271 



If we consider this principle of the identity of series, whose 

 terms within a finite distance from the origin are identical, as 

 established, we shall experience no difficulty in admitting the 

 perfect algebraical equivalence of such series, and their gene- 

 secure basis founded upon the general principles of analysis, and their truth 

 •was not, therefore, generally admitted amongst mathematicians. In the year 

 1798, Callet, the author of the logarithmic tables which go by his name, pre- 

 sented a memoir to the Institute for the purpose of showing that the sums 

 of such periodic series were really indeterminate : thus, if we divide 1 by 

 I + X and subsequently make x = 1, we get 



1 — 1+1 — 1 + &C. (1.) 



the value of which is — . In a similar manner, if we divide 1 + a; by 



1 + a; + a;2, we get for the quotient 



1 — x" + x3 — x^ + x^ — x^ + &c., 



which becomes the same series (1.), though the value of the generating func- 



2 

 tion under the same circumstances becomes — . The same remark applies 



to the result of the division of 1 + x + x^ + . . x"' by I + x + x- -\- . . x", 

 which produces the same series (1.) when a; =: 1, though under such circum- 



stances its generatmg function becomes — . 



This memoir of Callet gave occasion to a most elegant Report upon this 

 delicate point of analysis by Lagrange, who justified upon very simple prin- 

 ciples the conclusion of Daniel Bernoulli. The series which results from the 

 division of 1 + a; by 1 + .r + a^, if the deficient terms be replaced, becomes 



l+O.x — x^ + x^ + O.x'^— a^ + x^ + O.x'^ — x^ + &c., 



which degenerates, when x ■=■ \, into the series 



1+0—1 + 1+0—1 + 1 + &c., 



and not into the series (1.). The same remark applies to the series which 

 arises from the division of 1 + a; + . . a;*" by 1 + a; + . . . . x", n "Z m; 

 which becomes, when x z= I, 



1+0 + + + &c. — 1+0 + + &c. +1+0 + &c„ 



•which is equal, by Bernoulli's rule, to — . 



n 



But it is not necessary to resort to this expedient for the purpose of deter- 

 mining the sums of such series ; for the series 



Oi + On X + 03 a- + . . a x^~ + a, a'' + &c. 



is a recurring series resulting from the developement of 



a, + Oo a; + 03 x- -\- . . a x^" 



1 - xP 

 which becomes — when a: = 1. If we replace x by —, this fraction will 

 become 



