A Treatise on the Theory of Screws

36 THE THEORY OF SCREWS. [36- work done by the unit wrench on a in a twist of amplitude &>/ about the screw ®i is awf-sTaj, but this must be equal to the work done in the same twist by a wrench of intensity a, on the screw whence ip^Wi = ^al or oti = — . Pl Thus, to compute each co-ordinate an, it is only necessary to ascertain from the tables the virtual coefficient of a, and a>n and to divide this quantity by Pn- 37. The Virtual Coefficient of two screws may be expressed with great simplicity by the aid of screw co-ordinates. The components of a twist of amplitude a' are of amplitudes a'an ... a'a6. The components of a wrench of intensity ß" are of intensities ß"ßit ... ß"ß>. Comparing these expressions with § 32, we see that otn = ä otn, ßn = ß ßn, and we find that the expression for the work done in the twist about a, by the wrench on ß, is a'ß" [%Pi<*ißi + ... + 22>«a«/S<,]. The quantity inside the bracket is twice the virtual coefficient, whence we deduce the important expression =tp1a1ß1. Since a and ß enter symmetrically into this expression, we are again reminded of the reciprocal character of the virtual coefficient. 38. The Pitch of a screw is at once expressed in terms of its co- ordinates, for the virtual coefficient of two coincident screws being equal to the pitch, we have pa = 2p1a13. 39. Screw Reciprocal to five Screws. We can determine the co-ordinates of the single screw p, which is reciprocal to five given screws, a, ß, 7, 8, e. (§ 25.) The quantities p1; ... must satisfy the condition =0,