$ome co?iseque7ices that have been deduced from it. 1 a 



or rather 



00(1 +-)" — 00 =log(l +2;); .-. loga;=x (jr=° ~ 1), 



from which the exponential theorem, and thence the whole 

 theory of logarithms, may be readily derived. 



In Liouville's well-known memoir on General Differen- 

 tiation, in the thirteenth volume of the Journal de I' Ecole Pn- 

 lytechiiique, the distinguished author has, I think, fallen into 

 error, in consequence of being governed by the prevailing 

 views respecting the failure of the general form here discussed. 

 He is led (page 84-) to the formula 



which, in accordance with those views, he affirms (page 85)10 

 be " absolument fausse lorsque /; = 0." In justification of this, 

 he maintains that the definite integral 



r"^ dz 



Jo ' z 



is a finite quantity. 

 Now since 



•Vo -^=70 T-^°° + 7:2^-1:2:3-^ + ^^' 



and consequently the proposed integral, namely, 



Vo ~^"~7o T = -^^-^ + 1:2^-1:^^ + ^'- 



How this can be pronounced to be zero, I am at a loss to 

 conceive. That it is infinite, instead of zero, necessarily fol- 

 lows from its interpretation in the left-hand member of the 

 original equation, even if there were no internal evidence of 

 the fact. In the particular or extreme case considered, that 

 lett-hand member becomes a:°=l ; and the right hand mem- 

 ber is the series here exhibited multiplied by 0, the limiting 

 value of/;: the form therefore is merely a particular instance 

 of X 00 ; interpretable, as all the cases which this termi- 

 nates are interpretable, by the unambiguous form on the left. 

 Several eriors of like kind occur in Liouville's memoir; all 

 traceable to the same oversight respecting fundamental prin- 

 ciples. 



It may not be superfluous to remark, in reference to the 

 foregoing series for log (1-fx), that whenever that series is 



