48 
tral series. Thus—although the contrary has often been 
affirmed—we cannot legitimately infer from this theorem, 
without the aid of an additional principle, that 
1—14+1—1+4+1—1+4&e.=8. 
For, as already shewn, the series within the brackets at (13) 
is deficient by a quantity, which in this case is + 1. Intro- 
ducing this, (13) gives for s the ambiguous result $+ 3; 
that is, 1 or 0. The additional principle adverted to, and _ 
which is absolutely essential to the received conclusion, is that 
already stated at page 44; or, as Dr. Whewell briefly ex- 
presses it, ‘‘ that what is true wp to the limit, is true at the 
limit.” 
The differential theorem, therefore, can never be employed 
with success to sum either a divergent or a neutral series; or 
to convert either into a convergent series. 
There has been supposed to exist a perfect analogy between 
1—1+1—1+ &c., as the limiting case of _—g+g’—g’+ &c., 
and 1°—2? 4+ 3’—4° + &c., as the limiting case of 1—2°g + 
3°9?— 474° + &e., and that, in consequence of this analogy, 
we have as much right to affirm that 1?—2’?+ 3°—4’ + &e, 
is accurately expressed by 0, the limiting case of (1—g) 
(1+ )~, the fraction which generates 1° — 2°g + 3°g?—4°g*+ 
&e., as that | — 1+ 1— 1+ &c. is accurately expressed by 
z, the limiting case of , the fraction which generates 
1 
l+g9 
l~g+ g’—g' + &c. But there is a total absence of analogy 
between these two instances: the series 1 —g + g?— g’+ &e. 
presents a series of convergent cases from g = 0, uptog = 1; 
and whatever rule or formula enables us to find the summation 
in_all cases must necessarily enable us to find it in the extreme 
positive limits 0 and 1; for no values, short of those limits, 
can be the first and last of the admissible cases. But this rule 
or formula of summation, whatever it be, is constructed con- 
formably to certain hypotheses; viz. that the convergent 
