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National Advisory Committee for Aeronautics, Report - Elements of the Wing Section Theory and of the Wing Theory

The following paper, prepared for the National Advisory Committee for Aeronautics,
contains those results of the theory of Wings and of wing sections Which are of immediate
practical value. They are proven and demonstrated by the use of the simple conceptions of
“kinetic energy” and “momentum” only, familiar to every engineer; and not by introducing
“isogonal transformations” and “vortices,” which latter mathematical methods are not
essential to the theory and better are used only in papers intended for mathematicians and
special experts.
The method is more convenient on account of the greater simplicity of the problem, there
being one coordinate and one component of velocity less than with the three-dunensional flow.
But the two-dimensional potential is still a function of two variables and it represents a distri-
bution of velocity equivalent to a pair of functions of two variables. By means of introducing
the potential 9. great simplification of the problem has been accomplished, reducing the number
of functions to one. This simplification can now be carried on by also reducing the number
of variables to one, leaving only one function of one variable to be considered. This very
remarkable reduction is accomplished by the use of complex numbers.
The advantage of having to do with one function of one variable only is so great, and
moreover this function in practical cases becomes so much simpler than any of the functions
which it represents, that it pays to get acquainted with this method even if the student has
never occupied himself with complex numbers before. The matter is simple and can be
explained in a few words.