38 



trigonometrical series, arising from differentiating convergent 

 forms, are to be understood. 



IV. — 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 

 sura. 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" : 



yx-'^dx zz — a;-', \ x-^dx — or' —b~\ which is finite ; 



r»o nm f» + m 2 

 \ x--dx = 4- GO, \ x-^dx = + 00, \ x-^dx =. . 



J-m Jo J-m in 



If, then, we construct the curve whose equation \% y ■=. xr^, 

 and if OA =.—m, OB = + 7n, 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 



once from P to Q, we find, as above, , a negative result, 



m 



for the sura of two positive infinite quantities. The integral 

 then, y being infinite between the liraits, takes an algebraic 

 character, standing in much the same relation to the required 

 arithmetical result, which must have been observed in diver- 



