Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/334

the propeller remains contant (which is our fundamental condition), it is a matter of simple geometry to show that the area of the thrust grading curve (of which $w$ is the ordinate) must remain constant; consequently, if we suppose any small variation to take place, and represent the various quantities as given on Fig. 132, we have, for the condition of minimum expenditure of energy—

$\Delta \ r_{1}\times y_{1}=\Delta \ r_{2}\times y_{2},$

or

${\frac {\Delta \ r_{1}}{\Delta \ r_{2}}}={\frac {y_{1}}{y_{2}}}={\frac {w_{2}}{E_{2}}}{\mbox{÷}}{\frac {w_{1}}{E_{1}}}={\frac {w_{2}\ E_{1}}{w_{1}\ E_{2}}}.$

But, since the area of the load grading curve is constant, we have

$\Delta \ r_{1}\ w_{1}=\Delta \ r_{2}\ w_{2}.$

that is

${\frac {\Delta \ r_{1}}{\Delta \ r_{2}}}={\frac {w_{2}}{w_{1}}},$

so that we have—

${\frac {w_{2}\ E_{1}}{w_{1}\ E_{2}}}={\frac {w_{2}}{w_{1}}},$  or  ${\frac {E_{1}}{E_{2}}}=1,$

or

$E_{1}=E_{2},$

proving that for the conditions of greatest economy, the blade limits are points of equal efficiency.

Hence, although different proportions may be chosen for the ratio $r_{1}/r_{2}$ (the inner and outer radial limits of the blades), there are appropriate conjugate values which are conducive to the maximum efficiency, and these values are determined by the points of equal efficiency on a curve plotted from the equation of efficiency (§ 204, Figs. 126, 127).

§ 213. The Thrust Grading Curve.—We know that the square-ended form of grading curve assumed in the preceding section is inadmissible, and in order that the principle of conjugate blade limits should apply strictly to a real propeller blade we must extend the demonstration to include other forms.

Let us suppose that we regard the thrust grading as made up of a number of small component distributions of load, each of which is strictly in accordance with the hypothesis (Fig. 133). Then we can approximate as closely as we please to a smooth

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