A Treatise on the Theory of Screws

418 THE THEORY OF SCREWS. [381- The equations to the axis of the screw are «Pi + ya>3 — zm, _ bp2 + za^ — xm. _ cp3 + ®co2 — ya^ ®i a>2 a>3 If x, y, z are all simultaneously zero, then Q'Pi bp2 ops ®i *), w3 ’ and these are, accordingly, the conditions that the instantaneous screw passes through the centre of gravity. With these substitutions the co-ordinates become Pi’?/'= (&2 — c2) ®2w3; paVs" = (c2 - a2) aw, psV” = (a2 - b2) a>1Cl>2, P2V2" = (b2 - C2) ®2®s; p4y" = (c2 - a2) «..«j; peV= (a2 - b2) &>10>2; remembering that p1 = + a; p2 = - a, &c., we have %" + %"= 0; %" + ^/'=0; <+j?" = 0; but these are the conditions that the pitch of ?, shall be infinite; in other words the restraining wrench is a couple, as should obviously be the case. From the equations already given, we can find the co-ordinates of the instantaneous screw in terms of those of the restraining screw. We have / abc (y" - y2") (y3" - Vi") (Vs" - Vfs") V 2 (b2 — c2) (c2 — a2) (a2 — b2) , TT b2— c2 and W1 = H-.-„---- ; «hh - % ) c2 — a2 w- b(V3"-Vi"y a>3 = H If we make a2-b2 C (vs' ~ Vs") ’ LO2 = ato,2 + ba2 + ca>32, h302 = apax2 + b3w22 + c3<a32, then we have 0^ = ‘Pi + pe La2 - h3\ < 2p, 2,abc ) ’ 02 = (O1 'P2 +Pe _ La2-h3\ < 2p2 2abc J ’ a _ (Pz+Pe , “3 — Ö>2 I 5--1- V 2ps Lb2 - h3\ 2abc ) 'Pt + Pe Lb2 — h3\ < 2p4 2abc ) ’ 04 = tu2 a fPs+Pe Lc2 — h3' 0s = w3- +........a , \ 2j>5 2abc , 0« = ö>3 Ps + Pe < %Ps Lc2 - h3\ 2abc ) ’ In these expressions, pe is the pitch of 0, and is, of course, an indeterminate quantity. 382. Remark on the General Case. If the freedom of a body be restricted, then any screw will be permanent, provided its restraining screw belong to the reciprocal »system. For the body