Page:Philosophical Transactions - Volume 145.djvu/188

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169
mr. w.h.l. russell on the theory of definite integrals.

Let , then we have

where is of course less than unity; an integral given by Abel.

When is less than unity we can always integrate with respect to , but may obtain a single integral more simply by proceeding as follows:—

We have

consequently we find by summing a geometrical progression,

When this result coincides with that last obtained. We may obtain a very general result by applying Fourier's theorem to the series of Lagrange and Laplace as follows:—

If , and ,

we have

Now we generally have
whence
and

Hence substituting in the above series, we find

Consequently we find the following definite integral: