Page:Philosophical Transactions - Volume 145.djvu/178

and we find from this the series

${\displaystyle x^{4}\left\{1+{\frac {\mu ^{2}}{{\frac {5}{2}}\cdot 1}}\cdot {\frac {x^{2}}{2^{2}}}+{\frac {\mu ^{4}}{{\frac {5}{2}}\cdot {\frac {7}{2}}\cdot 1\cdot 2}}\cdot {\frac {x^{4}}{2^{4}}}+\mathrm {\&c.} \right\}={\frac {3}{2\mu ^{3}}}\left\{\left(\mu x^{2}-x\right)\epsilon ^{\mu x}+\left(\mu x^{2}+x\right)\epsilon ^{-\mu x}\right\}}$

(I.)

Whence we find, putting ${\displaystyle \mu }$ for ${\textstyle {\frac {\mu ^{2}x^{2}}{2^{2}}}}$,

${\displaystyle {\begin{aligned}\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}d\theta &{\sqrt {\cos \theta }}\ \varepsilon ^{\mu \cos ^{2}\theta }\cos \left(\mu \sin \theta \cos \theta +{\frac {5\theta }{2}}-\tan \theta \right)\\&={\frac {\sqrt {\pi }}{2\mu ^{2}\varepsilon }}\left\{2\mu \varepsilon ^{2{\sqrt {\mu }}}-{\sqrt {\mu }}\ \varepsilon ^{2{\sqrt {\mu }}}+2\mu \varepsilon ^{-2{\sqrt {\mu }}}+\varepsilon ^{-2{\sqrt {\mu }}}\right\}\end{aligned}}}$

Next consider the symbolical equation

and assume as the transformed equation

Then

${\displaystyle u=(\mathrm {D} -2)v,}$

and

${\displaystyle v=\mathrm {C} _{1}x\varepsilon ^{\mu x}}$

where ${\displaystyle 1}$, ${\displaystyle \alpha }$, ${\displaystyle \beta }$ are the three cube roots of unity.

Hence

${\displaystyle }$

We must determine ${\displaystyle \mathrm {C} _{1}}$, ${\displaystyle \mathrm {C} _{2}}$, ${\displaystyle \mathrm {C} _{3}}$ according to the series we have to sum.

If

${\displaystyle }$

we find

Whence

${\displaystyle \int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}\!\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}d\theta d\varphi \varepsilon ^{\mu \cos \theta \cos \varphi \cos(\theta +\varphi )}\cos ^{-{\frac {1}{3}}}\theta \cos ^{\frac {1}{3}}\varphi }$