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National Advisory Committee for Aeronautics, Technical Notes - A Least Squares Procedure for the Solution of the Lifting Line Integral Equation

The distribution of lift over_the span of a wing.
in uniform motion is determined, according to the Prandtl
theory of the lifting line, as the solution of.a singu—
lar integro~differential equation the_mathematical com—
plexities of which are euch'that exact solutions have
been obtained only in very special cases. While several
methods have been devised for obtaining approximate solu—
tions to this equation, it is felt that a new procedure
based on a method of least squares which was presented
in reference 6 may be of practical interest.

In the usual procedures-an approximation to the lift
functiOn is assumed as the sum of a finite number_of ap~
propriate approximating_functions with undetermined coef—
ficients, after which the coefficients are determined in
various ways so that the 1ifting~line equation is approxi—
mately satisfied. While it might be expected that the
determination of these parameters would be most efficiently
accomplished by a method of least squares, the only ap~
plication of such a method known to the writer (referencez)
was not well adapted to numerical computation ,for arbi—
trarily varying chord an_d angle of attack. _In additi_on,
the single case treated was that of a. wing with discon—
tinuous angle of attack, for which the procedure as given
in reference 2 failed to give satisfactory results.

The purpose of the present paper is to presenttg.tcv
leastusquares precedure in which the major'part of the
numerical calculation can be readily carried out on a
computing ma_chine,_ and in which the amount- of'labor in—
volved is not dependent upon the nature of the variation
of the chord and the angle of attack. Since all the
previous- procedur.es are notably inadequate for the analy—
sis of wings with .discontinuous spanwise variation of '
angle of attack or_chord, an expl treatment of such
cases is included.