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equilibrium,” unless a restraining couple of sufficient magnitude be applied.^[1]

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

1. For a full exposition of the theory of this branch of the subject, reference should be made to Lamb's “Hydrodynamics,” Chap. VI., and numerous references therein cited; also “Nat. Phil.,” Thomson and Tait, 313, 320.

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