43 
extends beyond the series (11) to the right; a fact which is 
of no moment when this term merges in zero, but of infinite 
consequence when it merges in infinity. In such a case there- 
fore, a numerical error, of infinite amount, is committed at 
this step of the reasoning. Again, if the series within the 
brackets at (13), have its terms, like those of the original, 
tending to infinity, another numerical error of infinite amount 
comes to be introduced; and so on. In fact, just as in the 
method of definite integrals, before discussed, it is assumed, at 
each step of the reasoning, that terms infinitely great are ex- 
eluded; and not only so, but that the terms ultimately dimi- 
nish to zero. In the contrary case, therefore, the differential 
theorem is altogether inapplicable, leading to results which 
are equally inadmissible, whether the terms of the series in- 
crease without limit, or remain stationary in value: forming 
what has been called a neutral series, In this latter case the 
error committed will be finite; in the former it will be infinite. 
That an error is really committed in the application of this 
theorem to neutral series, will be more explicitly shewn pre- 
sently. 
Notwithstanding the imperfections noticed above, it should 
ereate no surprise that, in the applications of the differential 
theorem to particular diverging series, we so often obtain the 
algebraic function whose development really gives rise to the 
series, although no numerical approximation to the diverging 
series itself. ‘The function, whose development gives rise to 
the series, being represented by f(z), the series itself may be 
represented by f(x) — (x), agreeably to what has already 
__ been shewn in the former part of this Paper: it is the neglect 
of the function $(), in the particular application considered, 
that introduces the infinite numerical error into (13) ; leading 
us to conclude that, for the proposed value of x, f(x) =s, in- 
_ stead of f(x)—9(v)=s. Nowif there exist a convergent case 
of s, that is a case in which ¢(z) = 0, the differential theorem 
will compute it, furnishing the proper function of «x, f(.), 
E2 
