A Treatise on the Theory of Screws

10°] THE POTENTIAL. 89 If a free or constrained rigid body be displaced from a position of equi- librium by twists of small amplitudes, 0.',... 0n', about n co-reciprocal screws of leference, and if V denote the work done in producing this movement, then the reduced wrench has, for components on the screws of reference, wrenches of which the intensities are found by dividing twice the pitch of the corresponding reference screw into the differential coefficient of K with respect to the corresponding amplitude, and changing the sign of the quotient. It is here interesting to notice that the co-ordinates of the reduced impulsive wrench referred to the principal screws of inertia, which would gi ve the body a kinetic energy T on the screw 0, are proportional to 1 d? J dT 2P1 dø.”" 2pn d0n 100. Conjugate Screws of the Potential. Suppose that a twist about a screw 0 evokes a wrench on a screw y, while a twist about a screw </> evokes a wrench on a screw £ If 0 be reci- procal to then must </> be reciprocal to y. This will now be proved. The condition that 0 and f are reciprocal is PiØi^i + ... + pn0n£n = 0; but the intensities (or amplitudes) of the components of a wrench (or twist) are proportional to the co-ordinates of the screw on which the wrench (or twist) acts, whence the last equation may be written Pi0. + • • • + pn0n'^n" = 0 ; but we have seen (§ 99) that //__ . 1 d o it__________ . 1 dV^ ~ 2P1 dtf ’ • ’ • U ~ + 2pn : whence the condition that 0 and f are reciprocal is Now, as is an homogeneous function of the second order of the quantities </>/, ... we may write V,/, = Au<fy2 + ... + Ann<f>n - + 2j412<^>i </>2 + 2A13<f>1'<j)..' + ..., in which Ahk = A kh. Hence we obtain:— -,, * = 2 {.4!,</>/ + A..2(f>2 + ... + Aln(f>n'}. dyy