38 
trigonometrical series, arising from differentiating convergent 
forms, are to be understood. 
1V.—It remains now to be noticed that in some of the 
more advanced parts of analysis—especially in the doctrine of 
definite integrals—conclusions have been reached which seem 
to contradict the proposition endeavoured to be established in 
this Paper, viz. that convergent infinite series have no finite 
sum. But all such conclusions will be found upon examina- 
tion to originate in mistake. I proceed to examine the more 
important of these. 
The following has been recently offered, by a very cautious 
writer, in support of the statement that ‘*1+ 2+ 4+ &c. ad 
infinitum, is an algebraic representative of — 1, though it only 
gives the notion of infinity to any attempt to conceive its 
arithmetical value” : 
b 
\o-*da =—a2, \ adzx = a~ —b", which is finite ; 
a 
0 m +m 2 
\ a "dz = + ow, ) a’dzr = +, y reds 
—m 0 —m m 
If, then, we construct the curve whose equation is y = a~, 
and if OA =—m, OB = + Mm, we find the areas PAOY... 
and QOBY... both positive and infi- 
nite, which agrees with all our notions 
derived from the theory of curves. Again, | 
if we attempt to find the area PYQB, 
by summing PAOY and YOQB, we 
find an infinite and positive result, which 
still is strictly intelligible. But if we 
want to find the area by integrating at 
2 F 
once from P to Q, we find, as above, — —, a negative result, 
m 
for the sum of two positive infinite quantities. The integral 
then, y being infinite between the limits, takes an algebraic 
character, standing in much the same relation to the required 
arithmetical result, which must have been observed in diver- 
