462
MOSSOTTI ON THE FORCES WHICH REGULATE
The expression for
will then be reduced to
(5)′ |
![{\displaystyle {\begin{aligned}F&=4\pi f{\frac {1}{r}}\int _{0}^{r}(T_{0}e^{\alpha r^{\prime }}+V_{0}e^{-\alpha r^{\prime }})r'\,dr'\\&{}+4\pi f\int _{r}^{\infty }(T_{0}e^{\alpha r^{\prime }}+V_{0}e^{-\alpha r^{\prime }})dr'\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85b04c5e9e797e780a8fad906e3782619d01ba6d) | |
All the quantities
and
being null, except
and
, the values of
will also be null, except that of
: the formula (2)′ will then give
When
we must have
; we must then also have
, and there will remain only
.
By performing the integrations of the formula (5)′ within the limits indicated, and observing that
, we shall obtain
|
![{\displaystyle F=-k{\frac {V_{0}}{r}}\left(e^{-\alpha r}-1\right);}](https://wikimedia.org/api/rest_v1/media/math/render/svg/254f4cdd17853cca52462382f05b197b5a2e62b9) | |
As, in the differential expression for
, we may change
into
, and x into
, without any change taking place in its value, and as a similar change may be made in respect to the other coordinates, it follows that, by taking the point
,
,
, as the origin of the coordinates, we shall be able, in the two preceding formulas, to put
|
![{\displaystyle r={\sqrt {(x-\mathrm {x} ^{2}+(y-\mathrm {y} ^{2}+(z-\mathrm {z} ^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b70d8acf50b5b371073f95f2c101403c5c04eea) | |
or, generally,
|
![{\displaystyle r_{\nu }={\sqrt {(x-\mathrm {x} _{\nu })^{2}+(y-\mathrm {y} _{\nu })^{2}+(z-\mathrm {z} _{\nu })^{2}.}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46f2d0697b66bf2ff42141cc0000fec61c7d896d) | |
Now if, by placing the origin of the coordinates in the centre of each molecule respectively, we substitute these expressions of
and
, and that previously found for
in the equation (III)′, and take successively for
as many constants as there are molecules, we shall find that the equation
|
![{\displaystyle \Sigma kV_{0}^{(\nu )}{\frac {e^{-\alpha r_{\nu }}}{r_{\nu }}}=\Sigma kV_{0}^{(\nu )}{\frac {e^{-\alpha r_{\nu }}-1}{r_{\nu }}}+\Sigma {\frac {4\pi (g{\bar {\omega }}_{\nu }+f\mathrm {q} _{\nu })\delta _{\nu }^{3}}{3r_{\nu }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6db2f0c8c7b2d3a8f85412863d961e09fa9d4410) | |
will be satisfied by taking for each molecule
|
![{\displaystyle V_{0}^{(\nu )}={\frac {4\pi }{3k}}(g{\bar {\omega }}_{\nu }+f\mathrm {q} _{\nu })\delta _{\nu }^{3}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c7591d7ff3c3cbda5a00f91790f8173a66f29b9) | |
By substituting for
the value just found, we shall finally have
(IV)′ |
![{\displaystyle F=-{\frac {4\pi }{3}}\Sigma (g{\bar {\omega }}_{\nu }+f\mathrm {q} _{\nu })\delta _{\nu }^{3}{\frac {e^{-\alpha r_{\nu }}-1}{r_{\nu }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51efad4220a6e48f3cdb27198b86b516226141eb) | |