Page:Philosophical Transactions - Volume 145.djvu/190

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$\cos\left (\tfrac{\mu}{4}\cos\theta\cos\varphi\sin(\theta+\varphi)+\frac{\theta}{2}+\varphi-(\tan\theta+\tan\varphi)\right )$ $=\frac{4\sqrt{\pi}}{\varepsilon^2}\int^{\pi}_{-\pi}d\theta\varepsilon^{\alpha\cos\theta}\cos\alpha\sin\theta.\varepsilon^\frac{\mu\cos2\theta}{\alpha^2}\cos\frac{\mu\sin2\theta}{\alpha^2}-\frac{4\pi^{\frac{3}{2}}}{\varepsilon^2}$|undefined

But we may effect these reductions systematically by means of the following proposition due to :&mdash;

$=\frac{1}{2\pi}\int_{0}^{\pi}d\theta\{(\varphi_1(x\varepsilon^{i\theta})+\varphi_1(x\varepsilon^{-i\theta}))(\varphi_2(\varepsilon^{i\theta})+\varphi_2(\varepsilon^{-i\theta}))\}.$

has also proved in the same paper, that if the sums of the three series $\begin{align} & a_0+a_1\ x+a_2\ x^2+a_3\ x^3+\mathrm{\&c.}\\ & b_0+b_1\ x+b_2\ x^2+b_3\ x^3+\mathrm{\&c.}\\ & c_0+c_1\ x+c_2\ x^2+c_3\ x^3+\mathrm{\&c.}\end{align}$ are known, we may determine the sum of the series

by means of a double integral, but we shall not want this in what follows.