A Treatise on the Theory of Screws

378] THE THEORY OF PERMANENT SCREWS. 415 a — 62) + Y(ßs + ö6) — Z (03 + 04), b(é3-éi) + z (é1 + é3)-x^r, + ø3), C(ø5-ø8) + X(& + ø4)- Y(é1 + é2). Before substitution for X, Y, Z it will be convenient to use certain abbre- viations, 0i — 02 = 61; — 03 = Pl > 01 + 0'2 — \1, 01 + 02 = > 0S' — Ø4 = e2; Ø3 — Ø4 = p2~, Ø3 + Ø4 = Ø3 + Ø4 = "a, 0/-06' = e3; 05-06 = Psi 0s' + 06=K, 0s + 06=a>3. With these substitutions in v2 the square of the velocity of the element we readily obtain after integration and a few reductions and taking the total mass as unity, %fv2dm = a20i2 + dß.2 + b202 + b202 + c2052 + c-Ø.r + abe2piws — ace3pi<i>2 — Xi®2®3 (&2 — c2) + bce3p2Wi - baeip2a>s — \m3^i (c2 — a2) + caeip3m.2 - cbe2p3coi — X3a>i(o2 (a? — ft2), whence we easily find dT d0i' = + acp3a>2 — abp:M.. — (b2 — c2) ®2®s- If 7)1', ...7)” be the co-ordinates of the restraining wrench, then, as shown in S 368, A_dT V1 ~ Pi dør whence we deduce the following fundamental expressions for the co- ordinates of the restraining wrench :— Piv” — — acpa^i + abp2a)3 + (b2 — c2) ft>2ö>3> P2V2' = + acp-iO>2 ~ abp2w3 + (b2 - c2) a)2a>3, P-iVs' = — od>pi<i>3 + cbp3a>i + (c2 — a2) ö>sWi, Pt")" = + abpiw3 — cbp3(i>i + (ca — a2) ®3«i > PiV" = ~ bcp2(Oi + acpio)2 + (a2 — b2) ö>j®2, PhVs" = + bcp3wi - acpiw2 + (a2 - b2) As usual, we here write for symmetry p1 = -2ra-, p3 — — a', p3 = + b-, Pt = — b; p3 = + c; p3 = -c. We verify at once that PiVi'Øi + ■ • • PtV^Øa = 0, but this is of course known otherwise to be true, because the restraining screw must be reciprocal to the instantaneous screw.