Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/136

§ 89 equilibrium,” unless a restraining couple of sufficient magnitude be applied.

The third category possesses a particular interest in relation to aerial flight. The transverse force is characteristic of cyclic motion and is found as a consequence of the superposition of a cyclic motion on a translation, as in Fig. 48. It is due to the greater tension on the upper than the under surface of any circuit, such as that of the solid of substitution, $$\alpha\ \alpha\ \alpha$$; this difference of tension is indicated by the numerical superiority of the $$\psi\ \phi$$ squares in the region adjacent to the upper surface.

The connection between cyclic motion and a transverse force can be independently established by taking the transverse force as hypothesis and proving cyclic motion as a consequence.

§ 90. Transverse Force Dependent on Cyclic Motion — Proof.—Let $$A\ B$$ (Fig. 51) be successive positions of the body or filament at the beginning and end of a short interval of time, to which the transverse force is applied. Let it be granted that the filament exert a force $$F$$ on the fluid at right angles to its direction of translation, and let us suppose that this force be sustained by a distributed system of forces, $$f\; f\; f_1\ f_1$$, etc., acting from the boundary of the region, and let the line $$S\ S$$ represent the mean position of the force $$F$$ during the period under consideration.

Now the force $$F$$ forms with the forces $$f\ f$$ and $$f_1\ f_1$$ two couples (which from considerations of symmetry may be taken as equal) of opposite sign, that to the right being counterclockwise and that to the left clockwise. Assuming a steady state, the first of these is continuously engaging with and acting on undisturbed fluid on the right of the line $$S\ S$$, and must therefore be communicating to it counterclockwise angular momentum, and the following couple must be communicating clockwise angular