are supposing that the ah- that reaches the plane
between
and
has received momentum pro rata with that within the region bounded by
that is to say:—
|
![{\displaystyle M_{\alpha }\left/{\frac {M_{1}}{2}}=n\right/2\ \alpha \ \kappa ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/094b14dfe53f17d4aefd97d90b579b2394b89090) |
|
or |
![{\displaystyle M_{1}=M^{\alpha }\ {\frac {4\ \alpha \ \kappa }{n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f32ef85753ecbe9f8b7ed20c89648af14ad1678) |
|
that is, by (1) |
![{\displaystyle \epsilon ={\frac {M_{\alpha }}{M_{\alpha }\ \left(1+{\frac {4\ \alpha \ \kappa }{n}}\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2730069f75fdf649256265cca6de38e17f1e9a7) |
|
or |
![{\displaystyle \epsilon ={\frac {n}{n+4\ \alpha \ \kappa }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b762338f66d2c2ece5fdb4c33fb16a4498e9b116) |
|
We may take a constant to represent and our expression is—
|
|
![{\displaystyle \epsilon ={\frac {n}{n+e\ \kappa }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c34ed50d776670e98057efbc956aa59726d2912) |
|
§ 180.
and
, Plausible Values.—We are now able to find an expression for
in terms of
and
, for we already have the equation—
|
∴ ![{\displaystyle \quad {\frac {n}{n+e\ \kappa }}={\frac {c\ C}{\kappa }}-1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65430713d8aeeecac3d840b6f16668b839a13358) |
|
or |
![{\displaystyle c\ C={\frac {2\ \kappa \ n+e\ \kappa ^{2}}{n+e\ \kappa }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3b46923f1cec7fe640bf40cbf620dbaebed96c1) |
|
∴ |
![{\displaystyle c\ C\ n+c\ C\ e\ \kappa =2\ \kappa \ n+e\ \kappa ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5a0960e44a6a290cd05cfdffc88099a276cca53) |
|
∴ |
![{\displaystyle e\ \kappa ^{2}+2\ \kappa \ n-c\ C\ e\ \kappa =c\ C\ n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/855ee7c5ce04ab57f91b3c81ac5df024feb7d34d) |
|
or, |
|
whence—
|
|
![{\displaystyle {\boldsymbol {-}}\ {\frac {n}{e}}+{\frac {c\ C}{2}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74c7201fbb0af3f9892deb7473f540a9d483b2f9) |
|
the rest is a matter of choosing such a value of
as fits in best with experience. The author has taken
and this is the basis on which the following Table of plausible values of
and
is founded.
§ 181. Best Value of
Least Value of
—Assuming the values given in the Tables (I., II. and III.), we are now in