250 Mr. Babbage’s new methods of investigating the 
communicated these anomalous results, by following a very 
different course, arrived at several general theorems, which, 
when applied to the series I had obtained, gave the same 
results. This coincidence at first increased my confidence 
in the values so discovered, and I continued to examine the 
reason why my own formulas were in some cases defective. 
Mr. Herschei/s method was published in the Philosophical 
Transactions for 1814; and it was not until some time after 
that I perceived, that although the investigations were very 
different, the fundamental principle was the same in both 
methods. This induced me to attempt summing the same 
series by a direct process, and I succeeded in obtaining their 
sums by integration relative to finite differences, aided by 
certain peculiar artifices. The results obtained by this new 
plan, which is the first treated of in this paper, coincided with 
those already found, and seemed to confirm their truth, with- 
out in the least indicating the cause of the error : this cause 
however I now began to suspect, and, after some enquiry, 
I was at length able to detect. I have found that the method 
of expanding horizontally and summing vertically, will always 
lead to correct results, provided a certain series which I have 
pointed out, is finite. I have also shown how to express this 
series by a definite integral; and when this integral or this 
series has a finite value, the method may be depended on. 
In case this series or this definite integral is not finite, then 
the value of the series* multiplied by zero, must be added to 
* The investigation of this series is generally a task of considerable difficulty. I 
have however given an example, wherein the correction thus found, added to the 
sum indicated by the method we are considering, gives the true value of the series, 
which in this case is one whose sum has been found by Euler. 
