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 



{a -X)- zza- — - ^^3 - ^^^^^- 2_46,8a' ~ 



for all arithmetical values of «; 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 x 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 is in the supplementary correction 

 under the " &c." that such quantity must occur, when in that 

 correction a value greater than a^ is given to x. 



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. 



III. — Let us now pass to the consideration of other classes 

 of diverging series. 



There are two ways of investigating the difi'erential of 

 sin X, or of sin mx : 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 



