Page:Aerial Flight - Volume 2 - Aerodonetics - Frederick Lanchester - 1908.djvu/385

Appendix IV

Solution of the Cubic Equation

 

Equation (14), where cos θ=1, i.e. for the highest and lowest points of a phugoid, is,

$C={\sqrt {H}}\left(1-{\frac {H}{3H_{n}}}\right)$

in which C is known and it is required to find the value of H;

or

${\frac {C}{\sqrt {H}}}=1-{\frac {H}{3H_{n}}}$

$\therefore {\frac {C}{\sqrt {\left(3H_{n}{\frac {H}{3H_{n}}}\right)}}}=1-{\frac {H}{3H_{n}}}$

Let ${\frac {H}{3H_{n}}}=x$; the equation becomes

${\frac {C}{\sqrt {3H_{n}x}}}=1-x$

which can be solved on a slide rule of the "Gravet;" type as follows:—

Using the ordinary nomenclature for the four scales,

${\begin{cases}A\\B\\C\\D\\\end{cases}}$

(1) Set the cursor to C on scale D.

(2) Set 3 H_nn on scale B to C², as indicated on scale A.

(3) Move cursor to 1 on scale B (this divides C² by 3 H_n).

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