∴
n
a
2
V
2
=
m
1
V
2
{\displaystyle \quad n\ a_{2}\ V^{2}=m\ {\frac {1}{V^{2}}}\quad }
a
2
V
4
=
m
n
{\displaystyle \quad a_{2}\ V^{4}={\frac {m}{n}}}
(1)
When,
V
=
V
1
,
x
1
+
x
2
=
y
,
{\displaystyle V=V_{1},\quad \quad x_{1}+x_{2}=y,}
∴
n
(
a
1
+
a
2
)
V
1
2
=
m
1
2
{\displaystyle n\ (a_{1}+a_{2})\ V_{1}^{2}=m\ {\frac {1}{2}}\quad }
(
a
1
+
a
2
)
V
1
4
=
m
n
{\displaystyle \quad (a_{1}+a_{2})\ V_{1}^{4}={\frac {m}{n}}}
(2)
By (1) and (2) we have—
(
a
1
+
a
2
)
V
1
4
=
a
2
V
4
{\displaystyle (a_{1}+a_{2})\ V_{1}^{4}=a_{2}\ {\mbox{V}}^{4}}
that is,
V
1
4
V
4
=
a
2
(
a
1
+
a
2
)
=
x
2
(
x
1
+
x
2
)
{\displaystyle {\frac {V_{1}^{4}}{{\mbox{V}}^{4}}}={\frac {a_{2}}{(a_{1}+a_{2})}}={\frac {x_{2}}{(x_{1}+x_{2})}}}
or,
V
1
V
=
x
2
(
x
1
+
x
2
)
4
.
{\displaystyle {\frac {V_{1}}{\mbox{V}}}={\sqrt[{4}]{\frac {x_{2}}{(x_{1}+x_{2})}}}.}
The signification of this result is that if an aerodrome be designed to travel at a velocity
V
,
{\displaystyle {\mbox{V}},}
V
×
x
2
(
x
1
+
x
2
)
4
.
{\displaystyle {\mbox{V}}\times {\sqrt[{4}]{\frac {x_{2}}{(x_{1}+x_{2})}}}.}
As an example we may, as before, assign relative values
x
1
=
1
,
x
2
=
3
,
{\displaystyle x_{1}=1,\ x_{2}=3,}
V
1
=
3
4
×
V
4
=
.93
V
.
{\displaystyle V_{1}={\sqrt[{4}]{{\frac {3}{4}}\times {\mbox{V}}}}=.93\ {\mbox{V}}.}
If we take
x
1
=
x
2
{\displaystyle x_{1}=x_{2}}
V
1
=
1
2
4
×
V
=
.84
V
.
{\displaystyle V_{1}={\sqrt[{4}]{\frac {1}{2}}}\times {\mbox{V}}=.84\ {\mbox{V}}.}
Least Horse-power .—If we require to know the velocity of least power
V
2
{\displaystyle V_{2}}
V
2
=
V
1
3
1
4
=
.76
V
1
,
{\displaystyle V_{2}={\frac {V_{1}}{3^{\frac {1}{4}}}}=.76\ V_{1},}
V
{\displaystyle {\mbox{V}}}
V
2
=
V
×
x
2
3
(
x
1
+
x
2
)
4
.
{\displaystyle V_{2}={\mbox{V}}\times {\sqrt[{4}]{\frac {x_{2}}{3\ (x_{1}+x_{2})}}}.}
In the case of the values given above,
When
x
2
=
3
x
1
{\displaystyle \ x_{2}=3\ x_{1}}
V
2
=
.706
V
.
{\displaystyle \quad V_{2}=.706\ {\mbox{V}}.}
When
x
2
=
x
1
{\displaystyle \ x_{2}=x_{1}}
V
2
=
.638
V
.
{\displaystyle \quad V_{2}=.638\ {\mbox{V}}.}
![{\displaystyle {\frac {V_{1}}{\mbox{V}}}={\sqrt[{4}]{\frac {x_{2}}{(x_{1}+x_{2})}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/118d7029ebf93d5743912832b3d9e6ad2b8188ef)
![{\displaystyle {\mbox{V}}\times {\sqrt[{4}]{\frac {x_{2}}{(x_{1}+x_{2})}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3815819b68739bdd414d429aa6215c1f1ed8d44d)
![{\displaystyle V_{1}={\sqrt[{4}]{{\frac {3}{4}}\times {\mbox{V}}}}=.93\ {\mbox{V}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/351cf7c00861b6e831cb821cb7a835c31cf1b06f)
![{\displaystyle V_{1}={\sqrt[{4}]{\frac {1}{2}}}\times {\mbox{V}}=.84\ {\mbox{V}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd7edad9e20a2a11dd54c6f0f109958da3c5ad81)
![{\displaystyle V_{2}={\mbox{V}}\times {\sqrt[{4}]{\frac {x_{2}}{3\ (x_{1}+x_{2})}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f96e600b2934f7e33b9460a01630921aaa90491)
