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![{\displaystyle x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4) |
= direct resistance, of which—
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![{\displaystyle x_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308) |
is that due to body resistance, and
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![{\displaystyle x_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7af1b928f06e4c7e3e8ebfd60704656719bd766) |
is that due to aerofoil area.
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![{\displaystyle a_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf42ecda092975c9c69dae84e16182ba5fe2e07) |
= a normal plane area whose resistance is the equivalent of and
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![{\displaystyle a_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/270580da7333505d9b73697417d0543c43c98b9f) |
= a normal plane area whose resistance is the equivalent of
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![{\displaystyle V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845) |
= velocity of flight.
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![{\displaystyle \xi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b461aaf61091abd5d2c808931c48b8ff9647db) |
= coefficient of skin friction.
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and
are constants, as in preceding section, and
are further constants.
It can be shown that—
![{\displaystyle y=C_{1}\ {\frac {W^{2}}{L^{2}\ V^{2}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/711823ce59218642267131bd2980c65d8e2d0c05)
and
![{\displaystyle \ x=C_{2}\ a_{1}\ V^{2}+C_{3}\ \xi \ V^{2}\ L^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afd03168f589ea7f16f1a825996bb2ac2ac8a8aa)
We require to know the minimum value of
or, we require to solve for minimum value the expression—
![{\displaystyle C_{1}\ {\frac {W^{2}}{L^{2}\ V^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce2d2bda83f093e6e5e9086fc4682a2547ed1ed4)
![{\displaystyle \ +C_{2}\ a_{1}\ V^{2}+C_{3}\ \xi \ V^{2}\ L^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6be1f169495d5aea50581adbe2f2c9ba5b9b26b1)
Now
and
so that expression becomes
![{\displaystyle C_{1}\ {\frac {(W_{1}+k\ L_{q})^{2}}{L^{2}\ V^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70cdefca3d75b267415d72c73bccd95799db9c2f)
![{\displaystyle \ +C_{2}\ a_{1}\ V^{2}+C_{3}\ \xi \ V^{2}\ L^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6be1f169495d5aea50581adbe2f2c9ba5b9b26b1)
or
![{\displaystyle C_{1}\ W_{1}^{2}{\frac {1}{L^{2}\ V^{2}}}+C_{1}\ W_{1}\ 2\ k\ {\frac {1}{L^{2-q}\ V^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20a8a9a58833beef9c542d25f30605822004a483)
![{\displaystyle \ +C_{1}\ k^{2}\ {\frac {1}{L^{2-2q}\ V^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fc81f01ff3ffb15e8d1f902397c2360ab778976)
![{\displaystyle \ +C_{2}\ a_{1}\ V^{2}+C_{3}\ \xi \ V^{2}\ L^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6be1f169495d5aea50581adbe2f2c9ba5b9b26b1)
where
and
are variables.
Making a temporary substitution of constants in order to abbreviate, we have—
![{\displaystyle {\frac {a}{L^{2}\ V^{2}}}+{\frac {b}{L^{2-q}\ V^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/229ab65a6b6d496385f4d6fd896a8b65e3c4cdbb)
![{\displaystyle \ +{\frac {c}{L^{2-2q}\ V^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17e18a8330c2e813561cd85ba09a6072bda28c38)
![{\displaystyle \ +c\ V^{2}+f\ V^{2}\ L^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e745a9180c79387a75f48e520ac1a3eb8090239)
Differentiating in respect of
and
gives simultaneous equations as follows:—
![{\displaystyle {\boldsymbol {-}}2\ {\frac {a}{L^{3}\ V^{2}}}\ {\boldsymbol {-}}\ (2\ {\boldsymbol {-}}\ q)\ {\frac {b}{L^{3-q}\ V^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb5feddd1b16ddb33bc2690ef7a259df19fbd2a) ![{\displaystyle \ {\boldsymbol {-}}\ (2\ {\boldsymbol {-}}\ 2q)\ {\frac {c}{L^{3-2q}\ V^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0a1933e9e7bff53fefccd9637cf0b7d7195c146)
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(1)
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