The aerodynamic theory of the lifting line was first solved by Multhopp1. This solution uses a spanwise distribution of the control points accordingly a sinus-function and is applicable to unswept wings. For the bell-shaped lift distribution a direct solution of the lifting line formula can be derived and is described here.

The integral for total lift can be written as (Truckenbrodt2): (Tru. 5.42)

For a symetrical lift distribution this formula can be transformed: with ; (Equ. 00-01)

If the spanwise control points are aranged along a "central-angle" similar to the Multhopp procedure, the following terms can be written:
 local lift load and spanwise station (Mul. 07)

This leads to the final integration and its solution: with (Equ. 00-02) (Equ. 00-03)

The total required geometrical twist is, accordingly to Truckenbrodt, the sum of effective and induced angle of attack . Angles are calculated in [rad.]: (Tru 7.64)

The local effective angle of attack can be found with the term: (Tru. 7.11 & 7.28)

This term is now used in Equation (Equ. 00-01 b): (Equ. 00-04)

The solution for the induced angle of attack is more complex. Truckenbrodt gives the induced angle of attack for unswept wings: with the circulation (Tru. 7.67 & 7.70)

Using the following terms, we can transform this integrartion in the formulation as given in Multhopp:   (Equ. 00-05) (Mul. 13)

This formula now has to be solved for the bell shaped lift distribution:  (Equ. 00-06) (Equ. 00-07)

For this the term is split up in several parts, which can be integrated seperately. The calculation is described in the Appendix A1 . The final solution is: or (Equ. 00-08)

This gives the required calculation method for the geometrical twist , which can be transformed, using equation (Equ. 00-03): (Equ. 00-09a) (Equ. 00-09b)