Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/78

§ 41 relation of any given geometrical form, the velocity being read to a scale varying according to the linear dimension of the body, so that the diameter or some other definite linear dimension of the body is some definite and constant multiple or submultiple of the scale unit employed. Thus, if the curve be plotted for a one foot diameter circular plane and one foot per second velocity is represented by one inch, then for a two foot circular plane a one foot per second velocity will be represented by two inches, that is to say, a given pressure will be developed at one-half the velocity.

This result is independent of the value of the index connecting $$R$$ and $$V ,$$ or generally of the indices relating $$R ,\ V_1 ,\ A_1$$ and $$V ,$$ of Equation (5); it would appear to be fundamental.

§ 42. Resistance - Linear Curve.—We may express the relationship of linear dimension and resistance directly in the form of a curve in which $$R$$ is given by the ordinates as before, and $$l$$ is represented by the abscissae, the curve being drawn for any given value of $$V .$$ Now we have in Equation (5) $$R = c\ v^q\ A^{\frac{r}{2}}\ V ,$$ which we may write in the form $$R = c\ v^q\ l^r\ V^r ,$$ where $$l$$ is a linear dimension on which $$A$$ depends; in this form the expression is symmetrical in respect of $$l$$ and $$V ,$$ and we have Equation (7) $$V = c\ \frac{\nu}{l}$$ also symmetrical with regard to these quantities, so that the form of the $$R\ l$$ curve will be identical with the $$R\ V$$ curve.

Thus let $$a,\ a,\ a,$$ (Fig. 28) represent the $$R\ V$$ curve for a body of a certain geometrical form which we will entitle $$(F)\ l$$ where $$l$$ is a linear dimension, then, for any value of $$l$$ assigned to the body,