A Treatise on the Theory of Screws

 92 THE THEORY OF SCREWS. [102, The parameter va may be contrasted with the parameter ua considered in § 89. Each is a linear magnitude, but the latter depends only upon the co-ordinates of a, and the distribution of the material of the rigid body. Both quantities may be contrasted with the pitch p„, which is also a linear magnitude, but depends on the screw, and neither on the rigid body nor the forces. If a body receive a twist of small amplitude a' about one of the principal screws of the potential, then the intensity of the wrench evoked on the same screw is (§ 99):— + 1 2pa da' ’ but we have just seen that V= Fv^a1, whence we have the following theorem:— If a body which has freedom of the nth order be displaced from aposition of equilibrium by a twist about a screw a, of which the co-ordinates with respect to the principal screws of the potential are alt... an, then a reduced wrench (§ 96) is evoked on a screw with co-ordinates proportional to t>j2 Vn p °11’ ”■ p an’ where ... , plt ... , are the values of the quantity v, and the pitch p, for the principal screws of the potential. We can now express with great simplicity the condition that two screws 0 and </> shall be conjugate screws of the potential. For, if f) be reciprocal to the screw whose co-ordinates are proportional to A. v" A. Pi Pn we have:— + • •. + 1>n0n(l>n = 0. The expression for the potential assumes the simple form Fv^ + ... + Fvn"a^. If the function V be proportional to the product of the pitch of the displacement screw and the square of the amplitude, then every displacement screw will coincide with the screw about which the wrench is evoked. 103. Form of the Potential. The n principal screws of the potential form a unique group, inasmuch as they are co-reciprocal, as well as being conjugate screws of the potential. They therefore fulfil n (n — 1) conditions, being the total number available in the selection of n screws in an n system.