sums of several classes of infinite series. 271 
several results which were evidently incorrect, soon convinced 
me that some limitation existed, of whose nature I was not 
aware: it was not until some years after, that I found out 
the cause of the fallacies which had perplexed me; and still 
more recently, I discovered that the series on whose sum their 
truth or falsehood depended, might be expressed by a definite 
integral. By applying the criterion, which I shall presently 
explain, we cut off a great variety of series whose sums are 
erroneously given by the method in question ; whether this 
criterion does not exclude some series whose sums are cor- 
rectly given, is a point which I do not consider yet completely 
decided ; the difficulties to which the application of acknow- 
ledged principles have in this instance conducted us, appear 
worthy of the attention of mathematicians. A more strict 
method might have been pursued in determining the sum of 
that part of the series which is neglected ; but this in general 
leads to such differential equations, as cannot afford us much 
assistance. I have, however, given one example of this 
method, and I have shown that when the part which had been 
neglected, as being apparently equal to zero (but which is in 
fact a finite quantity,) is added to the sum furnished by the 
method of expanding horizontally and summing vertically , the true 
value of the series results. This confirms the explanation I 
have given of the reason of the apparent failure of that 
method. / 
It will be sufficient to point out the cause which leads to 
error, and to determine the conditions on which its existence 
depends for one only of the series ; suitable modifications of 
the reasoning will readily suggest themselves for the others. 
I shall therefore, at present, consider the theorem (A). If 
