: 3T 
It is plain that, however far such a series as this be ex- 
extended, a supplementary correction is always necessary to 
complete the equation; which correction must be infinite in 
value if the series be infinitely extended : and the analytical 
considerations offered above fully accord with this statement, 
the contrary of which could never have been entertained had 
not analysis seemed to justify the strange conclusion. All 
that analysis really authorizes us in saying, in reference to the 
extreme cases here considered, is—as the French analysts ex- 
press it—that ‘‘ la méthode ordinaire est en défaute.” 
Having mentioned the name of Abel in connexion with 
this subject, it may not be out of place to notice here, that 
that distinguished genius seemed inclined to trace the erro- 
neous results above to another cause: ‘‘ On applique aux 
séries infinies toutes les opérations, comme si elles étaient 
finies ; mais cela est-il bien permis? Jecrois que non. Ou 
est il démontré qu’on obtient la différentielle d’un série infinie 
en en prenant la différentielle de chaque terme?” And he 
then adduces the result, 
$= cosxz — cos2a + cos3a — &e. 
which he pronounces to be ‘¢ résultat tout faux.’* 
But I submit that no such results of differentiation can 
ever be absurd, unless the absurdity attaches to one or more 
of the individual terms. 
In the former part of this paper the examination was re- 
stricted to those classes of diverging series which arise from 
the development of fractions into geometrical series, and from 
the expansion of a binomial: but it is plain that the reasonings, 
in reference to the former developments, equally apply to 
those which arise from any fraction : 3 ; and the reasoning, 
in reference to the latter, equally applies to any root or power 
of (x). And, in what is shewn above, we see how divergent 
* Euvres, tome ii. p. 268. 
