Page:Philosophical Transactions - Volume 145.djvu/197

culation of the electrical arrangement depends upon the value of the definite integral,

$\int _{0}^{\infty }{\frac {\sin cz.dz}{(\varepsilon ^{2\pi x}-1)(a+b\sin ^{2}cz)}},$

I mention this on account of its analogy with the definite integral

$\int _{0}^{\infty }{\frac {dz\sin 2\pi z}{(\varepsilon ^{2\pi }+2\varepsilon ^{\pi }\cos 2\pi z+1)^{2}(\varepsilon ^{2\pi x}-1)}}$

whose value is found above. The principles contained in this paper will enable us at once to find the sums of the series

$1+x+{\frac {x^{\frac {1}{2}}}{1.2}}+{\frac {x^{\frac {1}{3}}}{1.2.3}}+{\frac {x^{\frac {1}{4}}}{1.2.3.4}}+\mathrm {\&c} .$

$1+{\frac {\tan \theta }{1}}+{\frac {\tan 2\theta }{1.2}}+{\frac {\tan 3\theta }{1.2.3}}+\mathrm {\&c} .$

$1+{\frac {\sec \theta }{1}}+{\frac {\sec {\frac {\theta }{2}}}{1.2}}+{\frac {\sec {\frac {\theta }{3}}}{1.2.3}}+\mathrm {\&c} .$

and of many others which can be imagined, by means of definite integrals. The definite integral of Poisson given above occurs in the solution of a functional equation; and it is probable that series similar to those I have been discussing in this paper, may be useful in enabling us to express the solutions of other functional equations by definite integrals.