By William Frederick Durand

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**Read Online or Download Aerodynamic Theory: A General Review of Progress Under a Grant of the Guggenheim Fund for the Promotion of Aeronautics PDF**

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**Example text**

5), the quantities on the right will be known and we shall have a set of three simultaneous equations with a 1 , {31 and y1 as the unknowns. 2). 3). We may therefore conclude that it is always possible to write such expressions with four variables and with exponents such that the dimensions of the expression will be zero. 3) are of zero dimension. 7) 26 A IV. 4). 4). 3), the first affected by the exponent af~. 1), as representing any given term in the general equation of zero dimension. 7). These properties are either self evident, or the interested reader will readily supply a proof.

INTEGRATION OF PARTIAL DERIVATIVE EXPRESSIONS :; = 3 ax 2 y 2 If now we are given it is clear that to find the expression giving rise to this partial differential coefficient, we must simply reverse the process which gave it birth. We must integrate with reference to x and x alone. This will obviously give z =a x 3 y 2 If likewise we should be given oz - - = 2ax 3 y oy we shall naturally find the primitive of this expression in the same way by integrating solely with reference to y. Carrying this out, we have a xs y 2 as before.

This is the so-called II theorem which plays so important a part in many problems involving dimensions and kinematic similitude. 8). 3}, affected with suitable exponents as the individual term may require. It will be noted that the general form of the physical equation of zero dimension will contain unity as one of its terms, resulting from dividing through the general equation, not of zero dimension, by some one of its terms. This term unity, however, is, of course, of zero dimension and admits of representation by a term of the form (lit II~) simply by making A.