A Treatise on the Theory of Screws

393] THE THEORY OF PERMANENT SCREWS. 431 The permanent screw on this cylindroid will be one whereof the restrain- ing screw coincides with P. In general, the points corresponding homo- graphically to the points on the ray AB will form a ray CD. The inter- section 0, regarded as on CD, will be the correspondent of some point X on AB. The restraining screw corresponding to X will therefore lie at P, and will be provided by the constraints. Accordingly, X is a permanent screw on the cylindroid, and it is obvious from the construction that there can be no other screw of the same character. We can also deduce the expression for T in the two-system from the expression of the more general type in the three-system ; for we have T = T„ + (fi — p) + (v — X) 02030i 4- (X — /i) Ø/Øj^. Consider any screw on the cylindroid defined by 0/ = P02' + QØ3 , = PØ2 + QØ3; substituting, we obtain (Ø'é^ - 02'03) [(X - fi) P0.2 + (X - v) Q03], which we already know to be the form of the function in the case of the two- system (§ 384). 393. Freedom of the Fourth Order. The permanent screws in the case of a rigid body which lias freedom of the fourth order may be investigated in the following manner:—If a screw 0 be permanent, the corresponding restraining screw r/ must be provided by the reactions of the constraints. All the reactions in a case of freedom of the fourth order lie on the screws of a cylindroid. On a given cylindroid three possible if screws can be found. For, if we substitute ax + X/3,, a2 + \ß2, &c., for ifi, t]2, &c., in the equation 68 —c2 Th + Vi + c2-a2 773 + ^4 ft2 - Vs + ye _ Q a Vi-Vi 0 Vz-Vt c Vs - Ve