A Treatise on the Theory of Screws

122 THE THEOKY OF SCREWS. [134 We thus see that pt and p2 are linear functions of p/ and p2', the several coefficients A'B', A'B, &c., in these two equations being constant. The equation for w9s is thus to be transformed by a linear substitution for px and p2. Of course ue, being dependent only upon the position of X, is quite unaffected by the change of the screws of reference. We can therefore apply the well-known principle that the invariant of this binary quantic can only differ by a constant factor from the transformed value. The invariant is (X - w92) (p — ue2) - (p + Ug2 cos e)2. This must be true for every point X, and therefore for all values of It is necessary that the coefficients of the terms in the expression tig4 sin2 e - ue3 (X + v + 2/z. cos e) + Xv - /z,3 shall be severally proportional to those in the transformed expression Ug siU" t — Ug" (X p + 2^4 cos e) 4" Xu' — y'2. We thus obtain the two equations of condition, sin2 e X + p + 2/z' cos e' Xv — y1 sin2 e X + v + 2y cos e Xv — y2 The four quantities, X, y, p', e, may now be chosen arbitrarily, subject to these two equations, which are the necessary as well as the sufficient conditions. Indeed it is obvious that there must be but two independent quantities corresponding to the two positions of A' and B’. We may impose two conditions on the four quantities, and for our present purpose we shall make X = p ; y' = 0. The equations of X and e are then sin2 e' 2X X2 sin2 e X+ 2y cos e + v Xv — y2’ and we obtain x,= 2(Xi/-^) X + 2p cos e + p ’ ■ 2 ’ ■ 4 (Xv — /z2) s m- e = sm- e y---------——- ; (X + 2p cos e + v)2 X is thus uniquely determined, and the expression for sin-e' gives for e four values of the type + e, + (tt — e). The negative values are meaningless, and the two others are coincident, because the arc which subtends e on one side subtends tt — e' on the other. There is thus a single pair of screws of reference which permit the expression for to be exhibited in the canonical form c2Ug2 = X' (pf + p3-’).