indefinite velocity in an edgewise direction being superposed. It will be seen that in such a case as this the corresponding state of steady motion is unstable, unless a torque be supposed applied from without.
It is established that in the perfect fluid any body in steady motion, no matter what its form, experiences no resistance whatever in the direction of its flight, that is to say, the sum of the longitudinal components of pressure on its posterior surface is equal to the sum of those on the anterior surface. It is only in certain symmetrical cases, however, that the conditions of motion are stable without a force or forces applied to the body.
§ 89. Cases fall into Three Categories.—Taking the body and fluid as a combined system, cases fall naturally into three distinct categories: Firstly, those in which the fluid motion is in effect symmetrical, in which case the motion is in equilibrium without any applied force (this includes cases of unstable as well as cases of stable equilibrium). Secondly, cases in which the body is unsymmetrical and in which the motion involves the application of a couple or torque. Thirdly, cases in which cyclic motion is present and in which the motion involves a transverse force. Cases may occur which fall into both categories 2 and 3.
The first category has been sufficiently dealt with already in the present chapter and in Chap. I; the second is typified by the case of the inclined plane, and in a generalised form has been investigated by Kirchhoff, who has pointed out that there are three mutually perpendicular directions for any solid, in which, if it be set in motion and left to itself, the motion will continue indefinitely; in general it has been shown that one only of these directions is stable, the other two represent cases of unstable equilibrium. Generally speaking, a body having an aspect of greatest area such as an oblate spheroid, or a plane disc, tends to move “broadside on,” and if its motion at any time is disturbed it will oscillate about such natural “aspect of
