A Treatise on the Theory of Screws

[391- 430 THE THEORY OF SCREWS. + do + b (032 + 0i) yQ + (ßi — 0s2) (2 + 030i (pi - pi + ba — be) _+ 0ß2 (li - aaß - 0203 (l32 + cz0)_ + 03' r+ c(0i- + di) z0+(0i - 0i) H ’ + 0101 (pi - Pl + cb- ca) . + ((2 - by0) - 0-Å (li + CW,). The coefficients of 0i, 0i, 03' respectively each equated to zero will give three conics U= 0, V = 0, W = 0. These conics have three common points which are of course the three permanent screws. If we introduce a new quantity we can write the three equations (be - pi + fl) Øj + (li + cz0) 02 + (li - by0) 03 = 0, (li - cz0) 0i + (ac - pi + fl) 0., + (li 4- axi) 03 = 0, (li + by0) 0! + (li - ax(l) 02 + (ab - pi + fl) 03 = 0. The elimination of fl between each pair of these equations will produce the three equations U = 0, V= 0, W = 0. If therefore we eliminate 02, 03 from the three equations just written the resulting determinant gives a cubic for fl. The solution of this cubic will give three values for II which substituted in the three equations will enable the corresponding values of 0i, 02, 03 to be found. We thus express the co-ordinates of the three per- manent screws in terms of the nine co-ordinates of the rigid body and their determination is complete. It may be noted that the same permanent screws will be found for any- one of the systems of rigid bodies whose co-ordinates are 2/o. P2 + h> P*+h> pt+h, whatever h may be. 392. Case of Two Degrees of Freedom. We have already shown that there is a single permanent screw in every case where the rigid body has two degrees of freedom. We can demonstrate this in a different manner as a deduction from the case of the three- system. Consider a cylindroid in a three-system, that is of course a straight line in the plane representation (§ 200). Let this line be AB (fig. 44). If the movements of the body be limited to twists about the screws on the cylindroid, there may be reactions about the screw which corresponds to the pole P of this ray with respect to the pitch conic, in addition to the reactions of the three-system.