= Cross-resistance of an asymmetric planar waveform
{title2=$\delta=\Lambda^{-1}\int x_sy_s\,ds$}
An asymmetric periodic planar filament can have nonzero cross-tangent average $\delta$ despite periodic vertical position. Its translational resistance per arclength period is $c_\perp\Lambda[I+(\rho-1)M]$, with $M_{xx}=\beta$, $M_{xy}=\delta$ and $M_{yy}=1-\beta$. Full <force> balance can therefore yield transverse translation, so an axial-only propulsion formula needs a symmetry or constraint assumption.
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