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MR. W.H.L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS.

159

and we find from this the series

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

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

Next consider the symbolical equation

$(\mathrm {D} -1)(\mathrm {D} -3)(\mathrm {D} -5)u-\mu ^{3}\varepsilon ^{3\omega }u=0$ , where $\varepsilon ^{\omega }=x$ ;

and assume as the transformed equation

$(\mathrm {D} -1)(\mathrm {D} -2)(\mathrm {D} -3)v-\mu ^{3}\varepsilon ^{3\omega }v=0$ .

Then

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

and

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

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

Hence

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

If

we find

Whence

\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

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