^70 THIRD REPORT — 1833. 



s = a — {a — a + a — a + &c.) 



= a — * ; and therefore s = -r-*, 



2 



* The same principle would show that the equation 



*• = « +/(« +f(a+f{a + . . .))) 



is identical with the equation 



x= a +f (X) ; 

 and that 



* = «/(«/ (a/ (a...))) 

 is identical with 



X = af(x). 

 The example in the text is the most simple case of a class of periodic series, 

 the determination of whose sums to infinity has been the occasion of much 

 controversy and of many curious researches. The general property' of such 

 series is the perpetual recurrence of the same group of terms whose sum is 

 equal to zero : thus, if there should be p terms in each group, and if the num- 

 ber of terms n = mp + i, their sum would be identical with that of the i first 

 terms of the series ; and if we should denote those terms by aj, a.,, . . . o , 

 and if we should take the successive values of this sum for all the values of i 

 between 1 axidp inclusive, their aggregate value would be represented by 



pai + (p—1) a<^+ (p _ 2) a, + . . . a^. 

 of which the average (A) or mean would be represented by 



pay + (p — I) a.2 + (p — 2) as + . . . a^ 



P 

 If this periodic series was continued to infinity, it was contended by Daniel 

 Bernoulli, in memoirs in the 17th and 18th volumes of Novi Commmtarii 

 Petropolitani, for 1772 and 1773, that its sum would be correctly represented 

 by the average (A), in as much as it was equally probable that any one of 

 the p values would be the true one. Upon this principle it would follow, 

 that of the apparently identical series 



1 — 1 + 1 — 1 + 1 — &c. ... 



1+0— 1 + 1 + 0— 1 + 1 + &c. ... 



1-t-O + O— 1 + 1+0 + 0— 1+&C. 



12 3 



the first would be equal to — , the second to — -, and the third to — . In the 



A o 4 



same manner we should find 



1 + 1 — 1 — 1 + 1 + 1 — 1 — 1 +, &c. 

 equal to 1, and 



1+1+0 — 1 — 1 + 1 + 1+0— 1 — l+I + l +&C. 



equal to — -. The same observations would apply to the series 



1 + cos X + cos 2 X -\- cos 3 .r + cos 4 a; + &c. 

 and 



1 + cos a- + + cos 2 X -\- cos 3 .r + + cos 4 .r + &c. 

 where x is commensurable with 2 tt. 



These conclusions, however, though curious and probable, rested upon no 



