Page:Philosophical Transactions - Volume 145.djvu/181

162 We may extend this process, by performing operations with respect to the quantity ($$\mu$$). Thus we may operate on any of the integrals we have obtained by such a symbol as $\mathrm{F}\left (\frac{d}{d\mu}\right )$, where $$\mathrm{F}$$ is any rational function; and if it is an entire function, we have merely differentiations to perform. If it is a rational fraction, and the factors of the denominator are real and unequal, we may decompose it into simple rational fractions, each of which may, in its turn, be transformed into a simple integral. If we apply this operation to any of the results we have obtained, we immediately have a definite integral $$\iint..\mathrm{P}\varepsilon^{\mu \text{Q}}\text{F}(\text{Q})dv\ldots d\theta\ldots$$expressed in a series of single integrals, where the integrations are performed with respect to ($$\mu$$), and ($$\mu$$) may be taken between any limits. But ($$\mu$$) must in no case pass through zero, as the definite integrals, on which we operate with respect to ($$\mu$$), cannot be found for that value of $$\mu$$ by the processes we have been investigating. There are many other operations of a similar nature, which it is easy to imagine.

I am now come to the second part of this memoir, the investigation of those new methods of summation, and of the definite integrals corresponding to them, to which I have before alluded. Let us consider the series $1+\frac{x}{\beta}+\frac{x^2}{\beta(\beta+1).1.2}+\frac{x^3}{\beta(\beta+1)(\beta+2).1.2.3}+\mathrm{\&c},$ where ($\beta$ ) is an integer. The following integral is known:

Hence we find for the sum of the above series,

Next let us consider the same series when ($$\beta$$) is a fraction. We have

except for $$n=0$$, when

and we find for the sum of the series,