31 



a • ^*" t.' 



must connect to the correction — - — - : a correction 



1 — X I — ^ 



which is ambiguous as to sign, when x is negative. 



When a; is > 1, the series, omitting this correction, is oo ; 

 the correction itself is also oo, and opposite in sign : it is the 

 difference of these two infinites which is the finite undeveloped 

 expression. 



There is thus no discrepancy between a geometrical series 

 and the expression which generates it : nor is it the case that 

 by connecting the two by the sign of equality, we shall have 

 an equation algebraically true, but in certain cases arithme- 

 tically false, as has been frequently affirmed of late. The re- 

 verse of this affirmation is the more correct statement ; inas- 

 much as by interposing the sign of equality between y— ^ 

 and the series (1), instead of the series (5), we have an equa- 

 tion algebraically false, though, within certain limits, arith- 

 metically true : this last circumstance arising from the fact 

 that the omitted correction, which renders the equation alge- 

 braically defective, would have vanished of itself, between the 

 arithmetical limits adverted to, had it been introduced. Thus, 

 the series noticed at the commencement of this paper, viz. 



1+2-1-44-8 + 16 + &c., 



and which is intended to represent the development of r^-^j 



arises from expressing the general development of in the 



defective form 



1 + a; + a;^ -f- a;^ -f- cc* + . . . -i- «", 

 instead of, in the accurate form, 



\ ^x+x^-\-x^+x'-\ +x^ + j^, 



which defective form introduces arithmetical error only when 

 X exceeds unit. When x - % the error arising from this de- 

 fect is infinitely great ; the true form giving, in that case, 



