A Treatise on the Theory of Screws

CHAPTER VII . THE PRINCIPAL SCREWS OF INERTIA*. 78. Introduction. If a rigid body be free to rotate about a fixed point, then it is well known that an impulsive couple about an axis parallel to one of the principal axes which can be drawn through the point will make the body commence to rotate about that axis. Suppose that there was on one of the principal axes a screw y with a very small pitch, then a twisting motion about r) would closely resemble a simple rotation about the corresponding axis. An impul- sive wrench on y (i.e. a wrench of great intensity acting for a small time) will reduce to a couple when compounded with the necessary reaction of the fixed point. If wc now suppose the pitch of y to be evanescent, we may still assert that an impulsive wrench on y of very great intensity will cause the body, if previously quiescent, to commence to twist about y. We have stated a familiar property of the principal axes in this indirect manner, for the purpose of showing that it is merely an extreme case for a body with freedom of the third order of the following general theorem:— If a quiescent rigid body have freedom of the nth order, then n screws can always be found (but not generally more than ri), such that if the body receive an impulsive wrench on any one of these screws, the body will commence to twist about the same screw. These n screws are of great significance in the present method of studying Dynamics, and they may be termed the principal screws of inertia. In the present chapter we shall prove the general theorem just stated, while in the chapters on the special orders of freedom we shall show how the principal screws of inertia are to be determined for each case. * Philosophical Transactions, 1874, p. 27.