Page:Philosophical Transactions - Volume 145.djvu/188

Rh Let $$r=2$$, then we have

where $$(c)$$ is of course less than unity; an integral given by.

When $$2^r\alpha$$ is less than unity we can always integrate with respect to $$(z)$$, but may obtain a single integral more simply by proceeding as follows:&mdash;

consequently we find by summing a geometrical progression,

When $$r=2$$ this result coincides with that last obtained. We may obtain a very general result by applying theorem to the series of  and  as follows:&mdash;

If $$u=f(y)$$, and $$y=z+x\varphi(y)$$,

Hence substituting in the above series, we find

Consequently we find the following definite integral: