A Treatise on the Theory of Screws

229, 230] FREEDOM OF THE FIFTH ORDER. 247 To express this equation concisely we introduce two classes of subsidiary magnitudes. We write one magnitude of each class as a determinant. + piVi + ^iÄ-2/iOtj, «i, ßi, 7i p2ßi + Z2«2 — X2y2, + — a2> ßz> ?2 p3/33 + ^3a3-æ3V3> p^3 + ^ßi-y^, as. ßt, 7s PiÄ + ^A — «4^4, pjyt + ^ißt — y^t, «<> ßt, 7i pA + øsa5 - ®6y6, Psys + ^ßü-y^, as, ß5, y5 By cyclical interchange the two analogous functions Q and R are defined. pi“i + 2/171- ZA > Pißi + ^i-^y!, piVi + ^Ä-yi«!, ßi, Vi =L- p2a.2 + y2y2-zA> pA + x2y2, p2y2 + x2ß2- y2a2, ß2, y2 p3a3 + y3y3- z3ß3> psßs + ^-^ys, PsfY» + y&3, ß3> 7s P^i + ytyt-^ßt, PiÄ + ^«4-®474, ptyA^ßi-y^ ßt> 7* p3a5 + y5y5 — z3ß3, p3ßs + z6as-x6y6, psiy3 +x5ß3-ysa5, ß5, y5 By cyclical interchange the two analogous functions M and N are defined. The equation for p reduces to (P2 + Q- + R-~) P + PL + QM + RN = 0. The reduction of this equation to the first degree is an independent proof of the principle, that one screw, and only one, can be determined which is reciprocal to five given screws ; p being known, a, ß, y can be found, and also two linear equations between x, y', z, whence the reciprocal screw is completely determined. For the study of the screws representing a five-system we may take the first screw of a set of canonical coreciprocals to be the screw reciprocal to the system. Then the co-ordinates of a screw in the system are 0, 02> 03,...0O) while if X, p, v be the direction cosines of 0 and x, y, z a point thereon, and p6 the pitch we have (§ 43) (pfl + a) cos X - cos /z. + 2/ cos v = 0. We can obtain at once the relation between the direction and the pitch of the screw belonging to the system and passing through a fixed point. If Po = 0 and z and y be given, then the equation shows that the screw is limited to a plane (§ 110). 230. Six Screws Reciprocal to One Screw. When six screws, JlJ; ... Ae are reciprocal to a single screw T, a certain