Page:Philosophical Transactions - Volume 145.djvu/189

170 Again, from theorem, we have

These theorems of course suppose the series from whence they were derived to be convergent.

As examples we may take the following.

which we may modify thus; by eliminating $$(x)$$

Analogous methods apply to series involving numbers; thus we have

In this formula $(\alpha)$ must lie between 0 and 1, as it is necessary for the convergence of the above series that $$x$$ should be less than $$2\pi$$.

I now enter upon the consideration of the processes I have before mentioned for reducing multiple integrals to single ones. We easily see the truth of the following equation:&mdash;