A Treatise on the Theory of Screws

152] PLANK REPRESENTATION OF DYNAMICAL PROBLEMS. 141 These will evoke wrenches on E' and F' of the intensities HF HE e EF’ ’ EF’ respectively. But this pair of wrenches are to compound into a wrench of intensity h on H', and consequently we have KJ" HF 1 E’F’~eEF’ H'E' _ HE. ' E'F' EF ’ whence HF H'F’ HE : H'E' " If we take another pair of points, K and K', we have HF KF H'F' K'F' HE ' KE " H'E' : K'E” whence (HKFE) = (H’K'F'E'). Thus, the anharmonic ratio of any four points in one system is equal to that of their correspondents, and the two systems are homographic. The homographic axis intersects the circle in two points, which are the principal screws of the potential, i.e. a twist about either evokes a wrench on the same screw. Of course this homographic axis is distinct from that in § 139. But this homographic axis, like the former one, passes through the pole of the axis of pitch because the principal screws of the potential are i-eciprocal. 152. Work done by a Twist. Suppose that the body, when in equilibrium under the system of forces, receives a twist of small amplitude a' about any screw a, a quantity of work is expended, which we shall denote by Fva2a'2. In this, J1 is a constant, whose dimensions are a mass divided by the square of a time, and va is a linear magnitude specially appropriate to the screw a, and depending also upon the system of forces (§ 102). We may compare and contrast the three quantities,^, wa, va: each is a linear magnitude specially correlated to the screw a. The first and simplest, pa, is the pitch of the screw, and depends on the geometrical nature of the constraints; ua involves also the mass of the body, and the distribution of the mass relatively to a; va, still more complicated, depends also on the system of forces.