458
MOSSOTTI ON THE FORCES WHICH REGULATE.
`
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![{\displaystyle Q_{i}={\frac {1}{r}}Q_{i}^{(1)}-{\frac {1}{i}}{\frac {dQ_{i}^{(1)}}{dr}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d977e0fcd6a095ee338ce54b4e745609a0bd56d) | |
the last of these quantities will satisfy the equation (4), and will be its complete integral.
If the successive substitutions are performed, and, for brevity's sake, we make
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[1] | |
which gives, in the particular case of
,
, we shall have
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![{\displaystyle Q_{i}={\frac {a_{i}^{(0)}}{r^{i}}}Q_{i}^{(i)}+{\frac {a_{i}^{(1)}}{r^{i-1}}}{\frac {dQ_{i}^{(i)}}{dr}}+{\frac {a_{i}^{(2)}}{r^{i-2}}}{\frac {d^{2}Q_{i}^{(1)}}{dr^{2}}}\cdots \cdots \cdots +a_{i}^{(i)}{\frac {d^{i}Q_{i}^{(i)}}{dr^{i}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ac3653fcdfc186bdc1069a969a0524e51002f7d) | |
Now if we make
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![{\displaystyle {\begin{aligned}\Omega _{i}(r')&=\left\{{\frac {a_{i}^{(0)}}{r^{\prime i}}}+{\frac {a_{i}^{(i)}}{r^{\prime i-1}}}\alpha +{\frac {a^{(2)}}{r^{\prime i-1}}}\alpha ^{2}\cdots \cdots +{\frac {a_{i}^{(i)}}{r^{\prime 0}}}\alpha ^{i}\right\}e^{\alpha r'},\\\Omega '_{i}(r')&=\left\{{\frac {a_{i}^{(0)}}{r^{\prime i}}}-{\frac {a_{i}^{(1)}}{r^{\prime i-1}}}\alpha +{\frac {a_{i}^{(2)}}{r^{\prime i-2}}}\alpha ^{2}\cdots \cdots +{\frac {a_{i}^{(i)}}{r^{\prime 0}}}\alpha ^{i}\right\}e^{-\alpha r'},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a175a7a9033ffa483decf6cf468e759366100649) | |
where
is put instead of
we shall have
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![{\displaystyle Q_{i}^{'}=T_{i}^{'}\Omega _{i}(r')+V_{i}^{'}\Omega '(r'),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe06f78c9b2421ff9499507c63b310b263f29b47) | |
and the expression for
may take the form
(5) |
![{\displaystyle {\begin{aligned}F&=\Sigma _{0}^{\infty }\iint f\left\{{\frac {1}{r^{\prime n+1}}}\int _{0}^{r}\Sigma _{0}^{\infty }\Omega _{i}(r')T'_{i}r^{\prime n+1}\,dr'\right.\\&+\left.r^{n}\int _{0}^{\infty }\Sigma _{0}^{\infty }{\frac {\Omega _{i}(r')T'_{i}\,dr'}{r^{\prime n}}}\right\}P_{n}\sin \theta '\,d\theta '\,d\psi '\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08e85fcc330601bccc93520e260dd543a99e3aa8) | |
- ↑ The brackets are here employed in the same way as in Vandermonde's notation.