<Elliptic span loading> is a distribution of <lift> per unit span proportional to a half-ellipse, vanishing at both tips of a planar wing. For span $b$ and total <lift> $L$, write
$$
\boxed{L'(y)=\frac{4L}{\pi b}\sqrt{1-(2y/b)^2},\qquad -b/2\le y\le b/2.}
$$
The normalization follows by setting $s=2y/b$ and using $\int_{-1}^{1}\sqrt{1-s^2}\,ds=\pi/2$. It specifies a load distribution, rather than the wing outline. Classical <lifting-line theory> identifies this loading as the minimum-induced-drag solution for fixed total <lift> and span in its ideal planar setting.<Induced Drag Coefficient by NASA Glenn Research Center>[Lifting Line Theory, elliptic distribution and efficiency normalization]{ref}

The loading corresponds to <circulation> magnitude $\Gamma(y)=\Gamma_0\sqrt{1-(2y/b)^2}$, with $L'(y)=\rho U\Gamma(y)$ under the chosen positive-lift convention. Its ideal <downwash> at the lifting line is constant,
$$
\Gamma_0=\frac{4L}{\pi\rho Ub},\qquad
w=\frac{\Gamma_0}{2b},\qquad
D_i=\frac{Lw}{U}=\frac{2L^2}{\pi\rho U^2b^2}.
$$
Using dynamic <pressure> $q=\rho U^2/2$ recovers $D_i=L^2/(\pi qb^2)$ and the <span efficiency factor> $e=1$ in that normalization.

A <Fourier series> makes the minimum transparent. In the classical lifting-line expansion, put $y=(b/2)\cos\vartheta$ and
$$
\Gamma(\vartheta)=2bU\sum_{n\ge1}A_n\sin(n\vartheta),\qquad
C_L=\pi AR A_1,\qquad
C_{D_i}=\pi AR\sum_{n\ge1}nA_n^2.
$$
At fixed $C_L$ and $AR$, $A_1$ is fixed and every higher-order contribution to <drag> is nonnegative. The minimum has only $A_1$, yielding the half-ellipse. A <higher-harmonic penalty in span loading> quantifies departure from it: if $A_3=0.1A_1$ is the only additional term, $1/e=1+3(0.1)^2=1.03$, so $e\approx0.971$ and induced <drag> rises three percent under the stated constraints.

For an original loading example, $L=6000\,\mathrm N$ and $b=10\,\mathrm m$ give a central load $763.94\,\mathrm{N/m}$. At $y=b/4$ it is $0.8660$ of the central value. With $\rho=1.2\,\mathrm{kg/m^3}$ and $U=50\,\mathrm{m/s}$, the model gives $\Gamma_0\approx12.732\,\mathrm{m^2/s}$, $w\approx0.6366\,\mathrm{m/s}$ and $D_i\approx76.39\,\mathrm N$.

A proposed test integrates measured surface <pressure> around several wing sections to estimate $L'(y)$, then compares the normalized distribution and a wake survey with the model. An elliptic planform does not alone guarantee elliptic loading: section characteristics, incidence variation, deformation and operating conditions also enter. Nonplanar lifting systems, fixed bending-moment constraints and viscous or compressible effects change the optimization problem; the fixed-lift, fixed-span planar result should not be generalized beyond its assumptions.


 Elliptic span loading (source code)