172
mr. w.h.l. russell on the theory of definite integrals
${\begin{aligned}\therefore \ \int _{{\frac {\pi }{2}}}^{\frac {\pi }{2}}&\int _{{\frac {\pi }{2}}}^{\frac {\pi }{2}}\varepsilon ^{{\frac {\mu }{4}}\cos \theta \cos \varphi \cos \theta +\varphi }\cos ^{{\frac {3}{2}}}\theta \cos ^{1}\varphi \ d\theta d\varphi ,\\&\cos \left({\frac {\mu }{4}}\cos \theta \cos \varphi \sin(\theta +\varphi )+{\frac {\theta }{2}}+\varphi (\tan \theta +\tan \varphi )\right)\end{aligned}}$
(B.)
$={\frac {4{\sqrt {\pi }}}{\varepsilon ^{2}}}\int _{0}^{\pi }d\theta \left\{(\varepsilon ^{\cos {\frac {\theta }{2}}}+\varepsilon ^{\cos {\frac {\theta }{2}}})\cos \sin {\frac {\theta }{2}}\right\}\{\varepsilon ^{\mu \cos \theta }\cos(\mu \sin \theta )\}$
Hence we find, by comparing (A.) with (B.),
$\int _{0}^{\pi }d\theta \ \varepsilon ^{\mu \cos \theta }\cos(\mu \sin \theta )\left\{2\varepsilon ^{\frac {\cos 2\theta }{\mu }}\cos {\frac {\sin 2\theta }{\mu }}(\varepsilon ^{\cos {\frac {\theta }{2}}}+\varepsilon ^{\cos {\frac {\theta }{2}}})\cos \sin {\frac {\theta }{2}}\right\}=\pi .$
We have already proved that
$1+{\frac {2}{4}}x+{\frac {2.3}{4.5}}\cdot {\frac {x^{2}}{1.2}}+{\frac {2.3.4}{4.5.6}}\cdot {\frac {x^{3}}{1.2.3}}+\&c.$$={\frac {6}{x^{3}}}(x+2)+{\frac {6}{x^{3}}}(x2)\varepsilon ^{x}.$
Hence

$1+{\frac {3}{5}}\cdot {\frac {x}{2}}+{\frac {3.4}{5.6}}\cdot {\frac {x^{2}}{2.3}}\&c.$


$={\frac {12}{x^{4}}}(x+2)+{\frac {12}{x^{4}}}(x2)\varepsilon ^{x}{\frac {2}{x}},$
and

$1+\mu x+{\frac {\mu ^{2}x^{2}}{1.2}}+{\frac {\mu ^{3}x^{3}}{1.2.3}}+\&c.=\varepsilon ^{\mu x}.$


Consequently the theorem of M. Smaasen will give us the sum of the series
$1+{\frac {3}{2}}\cdot {\frac {\mu x}{5.1}}+{\frac {3.4}{2.3}}\cdot {\frac {\mu ^{2}x^{2}}{5.6.1.2}}+{\frac {3.4.5}{2.3.4}}\cdot {\frac {\mu ^{3}x^{3}}{5.6.7.1.2.3}}+\mathrm {\&c} .$
by means of a single integral, and we obtain
${\begin{aligned}\int _{{\frac {\pi }{2}}}^{\frac {\pi }{2}}\int _{{\frac {\pi }{2}}}^{\frac {\pi }{2}}&d\theta d\varphi \varepsilon ^{2\mu \cos \theta \cos \varphi \cos(\theta \varphi )}\cos ^{2}\theta \cos ^{3}\varphi \cos\{2\mu \cos \theta \cos \varphi \sin(\theta \varphi )+\tan \varphi 5\varphi \}\\&={\frac {\pi }{6\varepsilon }}\int _{0}^{\pi }d\theta \{6(\cos 3\theta +2\cos 4\theta )+6\varepsilon ^{\cos \theta }\cos(3\theta \sin \theta )\end{aligned}}$
$12\varepsilon ^{\cos \theta }\cos(4\theta \sin \theta )\cos \theta \}\varepsilon ^{\mu \cos \theta }\cos(\mu \sin \theta ).$
The fundamental idea of the preceding calculations, as will be readily seen, is as follows: to reduce every term of the series proposed to be summed by means of definite integrals to the form of the general term of the series whose sum is given by the common exponential theorem, and then to find the sum of the whole quantity contained under the signs of integration by means of that theorem. The factorials in the numerator of each term may be taken in any order we please relative to those of the denominator, provided that the same relative order is observed in every term throughout the whole series; moreover, we may use different integrals to express the