for the Sum of an Infinite Geometrical Series. 11 



sitive or negative, on account of the evanescence of #*". It 

 is usual to consider the infinite exponent in this expression 

 as invariable throughout all the changes of x within the limits 

 and 1 ; although it is known that for any fixed exponent 

 short of infinite, however great it may be, the expression into 

 which it enters becomes more and more considerable as x ad- 

 vances from towards 1 ; and notwithstanding the additional 

 fact, that when this exponent is actually infinite, the expres- 

 sion referred to becomes ultimately equal to — . 



But it is evident — due weight being given to the circum- 

 stances here mentioned — that this assumption, as to the inva- 

 riability of the infinite exponent, is unwarrantable and erro- 

 neous ; and that the exponent must follow some law of varia- 

 tion exactly fitted to counteract and neutralize the tendency 

 which, as x approaches to 1, the expression a' 00 " would other- 

 wise have to depart from zero, and ultimately to become — . 

 If x, at any stage of its approach to 1 , be generally repre- 

 sented by 1 — -r j then the law of variation alluded to will be 



expressed by go " = k go ' : that is, the exponent must vary as 

 k. For it is a remarkable fact that, commencing with the ex- 

 ponent 4* and proceeding onwards to infinity, we shall inva- 

 riably have 



©'-»-. (D---. (!)--. (?)'=-. 



/16801\ 16802 _ / 25684 ^ 25685 _ (y l X 



V16802/ -' 3,,, '-V25685/ -' 3 -» '" V 1 " / 



And since ('3 ...) co/ is necessarily zero, and no power 

 short of infinite can give zero, it follows that in order that 

 / i \«" 

 ( I — r- ) may be uniformly zero, and that all tendency to 



depart from zero may be counteracted, oo " must be k go ' ; so 

 that the strictly accurate form for S is 



•3... 



s = 



>+0-i) -0-t)' 



which is equal to — when k is infinite. And in this manner 



is the formula, employed in my paper (p. 363, last vol.), esta- 

 blished. 



