34 
two sides of the equation, and precluding the inconsistency so 
frequently affirmed to have place between them. 
From these simple considerations, it is easy to explain and 
reconcile such results as 
20) VAG 2 Aba | DAO wal 
for all arithmetical values of x; the *‘ &c.” being regarded as 
comprehending all that is necessary to render the second mem- 
ber of the equation a complete algebraical equivalent of the 
first. When 2 exceeds a’, the series becomes divergent ; and 
the first member of the equation becomes imaginary: and 
since it is impossible that any imaginary quantity can enter 
the series, it follows that it isin the supplementary correction 
under the “ &c.” that such quantity must occur, when in that 
correction a value greater than a’ is given to 2, 
From what has now be shewn, it may, I think, be legiti- 
mately inferred—as far, at least, as geometrical and binomial 
series are concerned— 
1. That whenever any such series becomes divergent for 
particular arithmetical values, what has been called above the 
supplementary correction becomes arithmetically effective, 
and cannot be disregarded without arithmetical error. 
2. And that so far from such series being, as usually 
affirmed, always algebraically true, though sometimes arith- 
metically false, on the contrary, they are always algebrai- 
cally false, though sometimes arithmetically true :—true in 
those cases, namely, and in those only, in which the proper 
algebraic correction becomes evanescent. 
I11.—Let us now pass to the consideration of other classes 
of diverging series. 
There are two ways of investigating the differential of 
sin a, or of sinmz: one by proceeding, as Lagrange has done, 
by actual algebraic development; and the other by employing 
the method of limits, independently of development. Accord- 
ing to Lagrange, we must proceed upon the assumption that 
Ee 
