438 Prof. J. R. Young on the Evaluation of 



'iftnit the convergent cases of the general forms considered 

 above — were proposed by Leibnitz and Daniel Bernoulli: 

 these methods have been the subject of a good deal of contro- 

 versy, which, as they are really true, would, no doubt, have 

 been spared if they had been based upon firm mathematical 

 principles. The considerations offered in the preceding part 

 of this paper will conduct to a very easy and satisfactory proof 

 of Bernoulli's rule. 



Let the terms of the limiting series in question be repre- 

 sented by /j, ^25 i^3> &c. These are such, that if we take the 

 first term, the sum of the first two, the sum of the first three, 

 and so on, we shall find that, after a certain term, the former 

 sums will recur ; so that the several results always return in 

 a certain definite cycle or period. Suppose this period of re- 

 sults to be 



oj ■■/"^nt'- Sj, 5.2, .% Smi i'grnJT.iaea to 



t^ithout stipulating anything as to the order in which 'these 

 m values succeed one another: let the sum o[ 7i terms of the 

 series be represented by S„, and put ^foc. hu J:-J 



S« = 5i, . f 



.,,1/ 



^/t "t* ^w+I + ^«+2 — S. 



3> 



^« I" tfi + i + tn + 2 T .... /n + TO— 1 — ^w« 



Now whichever of the individual results, in the entire period 

 of results, 5^ or 55, &c. be considered to represent, it is plain 

 that n may be chosen so as to render each of these equations 

 untrue; but notwithstanding this, it is equally plain that the 

 sum of them all will be perfectly accurate, whatever be 71 ; 

 that is, ♦ 



' . »0«-^ f^+l -1 — — t« + 2+ ••• = — — ■ -, 



riM\ -•/>• ^^ ^ ^ m ' 



a result which shows that whatever value we give to w, the 

 left-hand member of this equation is a constant quantity. But 

 if the series we are now discussing be the limiting case of a 

 general converging series proceeding according to the ascend- 

 ing powers of X, then, as already shown in the preceding part 

 of this paper, when n is infinite, 4 + 1, 4+^5 &c. are each zero. 

 Consequently 



q _^ » t 4- % -f ^ -f . . . Sm 



which is Bernoulli's rule. 



