abe 
nt Le oe 
31 
a : ax é 
must connect to a the correction — 1 a. a correction 
which is ambiguous as to sign, when z is negative. 
When z is > 1, the series, omitting this correction, is 0 ; 
the correction itself is also 0, 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 = 
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. 
Tear d 4-1 See 16 4- &e., 
_ and which is intended to represent the development of — 
arises from expressing the general development of _— in the 
—« 
defective form 
Ltatartaertaott+t...4+2°, 
instead of, in the accurate form, 
l+ata?+t+oetat+...+a2”" + —, 
which defective form introduces arithmetical error only when 
w exceeds unit. When x = 2, the error arising from this de- 
fect is infinitely great; the true form giving, in that case, 
