Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/264

weight of an aerodrome as constant in respect of the value of $A$, for the supporting members themselves must possess weight, and such weight must be some function of the area. This consideration will not affect the results given by props, i., ii., iii., for in these the hypothesis does not contemplate any change in the value of the “sail area” $A$.

We may regard the total weight $W$ of an aerodrome as consisting of two parts, $W_{1}$ and $W_{2},$ of which $W_{1}$ is constant being the weight of the essential load, and $W_{2}$ as the weight of the aerofoil which must vary in some way with the area $A,$ or $W_{2}=(F)\ A$ where the nature of the function must depend upon the conditions of design and construction.

Before any attempt can be made to investigate the influence of the matter under consideration, some assumption must be made as to the form of the function in question. The basis on which we have to found our assumption is that of some probable constructive method; thus we might suppose that as the aerofoil undergoes change of area its geometrical form is preserved and remains constant. If we take $L$ as representing the linear dimension of the aerofoil ($L$ may be chosen as any linear dimension so long as it is the same in all cases); then if the weight of the aerofoil per unit area were constant, we should have $W_{2}$ varies as $L^{2};$ or suppose we base our relationship on an assumption of constant geometrical form, but all scantlings of appropriate strength, investigation gives $W_{2}$ varies as $L.$ The actual relation, whatever it may be, depends upon the exigencies of design and can be established for any set of conditions empirically by designing aerofoils of different area and plotting an $L:W^{2}$ curve.

In detail we find that the weight of each element of the aerofoil structure may be represented by the simple expression $k\ L^{q}$ where $k$ and $q$ are constants which are different for the different functional elements. Now we know that an expression consisting of the sum of a number of quantities of the form $k_{1}\ L^{q_{1}},\ k_{2}\ L^{q_{2}},\ k_{3}\ L^{q_{3}},\ {\mbox{etc.}},$ may be approximated over a moderate range by a

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