# Page:Philosophical Transactions of the Royal Society A - Volume 184.djvu/48

48
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING.

${\displaystyle \int _{0}^{\pi }{\frac {\cos \phi \,d\phi }{\sqrt {r^{2}+c^{2}-2cr\sin \theta \,\cos \phi }}}}$

${\displaystyle ={\frac {2}{\mathrm {R} _{1}}}\int _{0}^{\frac {\pi }{2}}{\frac {\left(2\cos ^{2}\phi -1\right)\,d\phi }{\sqrt {1-{\frac {\mathrm {R} _{1}^{2}-\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\cos ^{2}\phi }}}}$

${\displaystyle ={\frac {2}{\mathrm {R} _{1}}}\int _{0}^{\frac {\pi }{2}}{\frac {2\left(\cos ^{2}\phi -{\frac {\mathrm {R} _{1}^{2}}{\mathrm {R} _{1}^{2}-\mathrm {R} ^{2}}}\right)-1+{\frac {2\mathrm {R} _{1}^{2}}{\mathrm {R} _{1}^{2}-\mathrm {R} ^{2}}}}{\sqrt {1-{\frac {\mathrm {R} _{1}^{2}-\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\cos ^{2}\phi }}}\,d\phi ,}$

${\displaystyle ={\frac {2}{\mathrm {R} _{1}}}\left\{{\frac {2}{1-{\frac {\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}}}\left[\int _{0}^{\frac {\pi }{2}}{\frac {d\phi }{\sqrt {1-{\frac {\mathrm {R} _{1}^{2}-\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\sin ^{2}\phi }}}-\int _{0}^{\frac {\pi }{2}}{\sqrt {1-{\frac {\mathrm {R} _{1}^{2}-\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\sin ^{2}\phi }}\,d\phi \right]-\int _{0}^{\frac {\pi }{2}}{\frac {d\phi }{\sqrt {1-{\frac {\mathrm {R} _{1}^{2}-\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\sin ^{2}\phi }}}\right\}}$

${\displaystyle ={\frac {4}{\mathrm {R} _{1}\left(1-{\frac {\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\right)}}\left[\left\{\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}+{\frac {1^{2}}{2^{2}}}{\frac {\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\left(\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}-{\frac {2}{1.2}}\right)+\,.\,.\,.\,\right\}-\left\{1+{\frac {1}{2}}{\frac {\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\left(\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}-{\frac {1}{1.2}}\right)+{\frac {1^{2}\cdot 3}{2^{2}\cdot 4}}{\frac {\mathrm {R} ^{4}}{\mathrm {R} _{1}^{4}}}\left(\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}-{\frac {2}{1.2}}-{\frac {2}{3.4}}\right)+\,.\,.\,.\,\right\}\right]}$${\displaystyle -{\frac {2}{\mathrm {R} ^{\prime }}}\left\{\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}+{\frac {1^{2}}{2^{2}}}{\frac {\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\left(\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}-{\frac {2}{1.2}}\right)+{\frac {1^{2}\cdot 3^{2}}{2^{2}\cdot 4^{2}}}{\frac {\mathrm {R} ^{4}}{\mathrm {R} _{1}^{4}}}\left(\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}-{\frac {2}{1.2}}-{\frac {2}{3.4}}\right)+\,.\,.\,.\,\right\}}$

[Cayley, 'Ell. Func.,' p. 54.]

${\displaystyle ={\frac {2}{\mathrm {R} _{1}}}\left\{\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}-2+{\frac {\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}{\frac {5\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}-7}{4}}+{\frac {\mathrm {R} ^{4}}{\mathrm {R} _{1}^{4}}}{\frac {162\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}-225}{128}}+etc.\right\}.}$

Now

${\displaystyle \mathrm {R} _{1}^{2}=4c^{2}-4c\mathrm {R} \cos \chi +\mathrm {R} ^{2}}$

${\displaystyle =4c^{2}\left(1-{\frac {\mathrm {R} }{c}}\cos \chi +{\frac {\mathrm {R} ^{2}}{4c^{2}}}\right)}$

${\displaystyle =4c^{2}\left(1-s\cos \chi +{\frac {s^{2}}{4}}\right)}$

Therefore

${\displaystyle \log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}=\log {\frac {8}{s}}-{\frac {s}{2}}\cos \chi -{\frac {s^{2}}{4}}{\frac {\cos 2\chi }{2}}-{\frac {s^{3}}{8}}{\frac {\cos 3\chi }{3}}-\,}$&c.

${\displaystyle =l+2-{\frac {s}{2}}\cos \chi -{\frac {s^{2}}{8}}\cos 2\chi -{\frac {s^{3}}{24}}\cos 3\chi -\,}$&c.

${\displaystyle {\frac {\mathrm {R} ^{2}}{4\mathrm {R} _{1}^{2}}}={\frac {s^{2}}{16}}+{\frac {s^{3}}{16}}\cos \chi +{\frac {s^{4}}{64}}\left(1+2\cos 2\chi \right)+\,}$&c.