A Treatise on the Theory of Screws

88 THE THEORY OF SCREWS. [98- by the principle of virtual velocities, the work done by small twists about the screws of reference must be zero, as far as the first power of small quantities is concerned. Finally, neglecting all terms above the second order, on account of their minuteness, we see that the function V, which expresses the potential energy of a small displacement from a position of equilibrium, is generally a homogeneous function of the second degree of the n co-ordinates, by which the displacement is defined. 99. The Wrench evoked by Displacement. When the body has been displaced to P, the forces no longer equilibrate. They have now a certain resultant wrench. We propose to determine, by the aid of the function V, the co-ordinates of this wrench, or, more strictly, the co-ordinates of the equivalent reduced wrench (§ 96) upon a screw of the system, by which the freedom of the body is defined. If, in making the displacement, work has been done by the agent which moved the body, then the equilibrium of the body was stable when in the position 0, so far as this displacement was concerned. Let the displacement screw be 0, and let a reduced wrench be evoked on a screw y of the system, while the intensities of the components on the screws of reference are % > ■ • • Pn ■ Suppose that the body be displaced from P to an excessively close position P', the co-ordinates of P', with respect to 0, being (§ 95) d.'+Sdf ...0n’+80n'. The potential V of the position P' is v+døisøi +- + de^e'1’ it being understood that 80.', ... 80f are infinitely small magnitudes of a higher order than 0.', ... 0n’. The work done in forcing the body to move from P to P' is V — V. This must be equal to the work done in the twists about the screws of reference whose amplitudes are 80.', ... 80f by the wrenches on the screws of reference whose intensities are y.'', ... yn". As the screws of reference are co-reciprocal, this work will be equal to (§ 33) + + ••■ + %yn'pn$0n. Since the expression just written must be equal to V - V for every position P in the immediate vicinity of P, we must have the coefficients of 80., ... 80n equal in the two expressions, whence we have n equations, of which the first is „ = 1 dV V1 + 2?1 dø.' • Hence, we deduce the following useful theorem:—