A Treatise on the Theory of Screws

102] THE POTENTIAL. 91 Introducing the value of V„, and remembering (§34) that a" = a!'ax and «/ = «'«!, we have the following n equations:— ai ^-‘4-11 “ ~z~ plj + «2-^12 + ... + anAln = 0, &c., &c. + a2An2 + ... + a„ p,^ = 0. From these linear equations alt... an can be eliminated, and we obtain a" an equation of the nth degree in . The values of substituted suc- cessively in the linear equations just written will determine the co-ordinates of the n principal screws of the potential. If the position of equilibrium be one which is stable for all displacements then Va must under all circumstances be positive. As it can be reduced to the sum of n squares all the roots of this equation will be real (§ 86) and consequently all the n principal screws of the potential will be real. Wc can now show that these n screws are co-reciprocal. It is evident, in the first place, that if $ be a principal screw of the potential, and if 6 be a displacement screw which evokes a wrench on y, the principle of § 100 asserts that, when 0 is reciprocal to then must also y be reciprocal to >S'. Let the n principal screws of the potential be denoted by Si, ... Sn, and let Tn be that screw of the screw system which is reciprocal to Si, ... Sn_i (§ 95), then if the body be displaced by a twist about 7'n, the wrench evoked must be on a screw reciprocal to ... Sn-P, but Tn is the only screw of the screw system possessing this property; therefore a twist about Tn must evoke a wrench on Tn, and therefore Tn must be a principal screw of the potential. But there are only n principal screws oi the potential, therefore Tn must coincide with Sn, and therefore Sn must be reciprocal to »S’j,... Sn_i. 102. Co-ordinates of the Wrench evoked by a Twist. The work done in giving the body a twist of small amplitude o about a screw a, may be denoted by Fvpa'2. In fact, remembering that aa^af... , and substituting these values for a/ in V (§ 100), we deduce the expression:— Fv* = -4iiai3 + • • • + Annan2 + 24^«, + + ... where F is independent of a and has for its dimensions a mass divided by the square of a time, and where va is a linear magnitude specially appropriate to each screw a, and depending upon the co-ordinates of a, and the constants in the f unction V (§98).