Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/262

§ 168 {| align="center" style="text-align: center; width: 100%"
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 * align="right" width="1%" | || width="1%" | $$\quad n\ a_2\ V^2 = m\ \frac{1}{V^2} \quad$$ or $$\quad a_2\ V^4 = \frac{m}{n}$$ || align="right" width="1%" | (1)
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 * align="left" | When, || $$V = V_1, \quad \quad x_1 + x_2 = y,$$ ||
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 * align="right" | || $$n\ (a_1 + a_2)\ V_1^2 = m\ \frac{1}{2} \quad$$ or $$\quad (a_1 + a_2)\ V_1^4 = \frac{m}{n}$$ || align="right" | (2)
 * align="left" colspan="3" | By (1) and (2) we have—
 * || $$(a_1 + a_2)\ V_1^4 = a_2\ \mbox{V}^4$$ ||
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 * align="left" | that is, || $$\frac{V_1^4}{\mbox{V}^4} = \frac{a_2}{(a_1 + a_2)} = \frac{x_2}{(x_1 + x_2)}$$ ||
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 * align="left" | or, || $$\frac{V_1}{\mbox{V}} = \sqrt[4]{\frac{x_2}{(x_1 + x_2)}} .$$ ||
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 * align="left" | or, || $$\frac{V_1}{\mbox{V}} = \sqrt[4]{\frac{x_2}{(x_1 + x_2)}} .$$ ||
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The signification of this result is that if an aerodrome be designed to travel at a velocity $$\mbox{V},$$ its “sail area” being such as will involve the least total resistance at that velocity, such an aerodrome will experience its least resistance when its velocity is reduced to—

As an example we may, as before, assign relative values $$x_1 = 1 ,\ x_2 = 3 ,$$ we have velocity of least resistance,

If we take $$x_1 = x_2$$ we shall have—

Least Horse-power.—If we require to know the velocity of least power $$V_2$$ we have by prop. iii.: $$V_2 = \frac{V_1}{3^{\frac{1}{4}}} = .76\ V_1,$$ or in terms of $$\mbox{V}$$ we have—

In the case of the values given above,