A Treatise on the Theory of Screws

390] THE THEORY OF PERMANENT SCREWS. 427 We thus find that when T is referred to the three permanent screws of the system, its expression must be T= afc + bfa + c082 + 2/ØÅ + 2^0,03 + 2ÅØÅ + {p,—v) øi'ø3ø3+(y—x) ø2ø3ø1 + (x—ø3ø1ø2. Let y" be the intensity of any wrench acting on a screw belonging to the system, and let represent the virtual coefficient between y and the first of the three screws of reference. Then, substituting for T in Lagrange’s equations, we have + aØj + h02 + g03 — (/z. — p) 020s = ^y", + /iöi + bø2 +yø3 — (v — x) ø3ø3 = 'us^r)", ++f^i + c^3 - (x - /*) ØÅ = vs^y". If y be the restraining screw, then an appropriate wrench y" should be capable of annihilating the acceleration, i. e. of rendering Ö! = 0 ; 02 = 0; 03 = 0; whence the position of y, and the intensity y" are indicated by the equations (y - p) 0203 = -sr^y", (X - v) 0301 — ^y", (p, - X) 0X03 = ^y”. We can now exhibit the nature of the correspondence between y and 0, for = PiVi + ^’la’Za + ^isVs, = ^1‘iVl + PzV2 + , - ^wVi + + PaVa- If we make H = 010203 -5- y', and omit the dots over 0X, &c., we have 0i (piVi + + W%) = S(y -ß), 02 (ywh + P2V2 + ^Va) = H (a - 7), 03 (™iaVi + +P3V3) = H(fi- a). We may reduce them to two homogeneous forms, viz. 0.L + 0,M+ 03N= 0, + ß0-,M + y03N = 0, where £ = 1^2 • = WÜere L ^dV1’ ^dy2’ idyi- 390. Geometrical Construction for the Permanent Screws. We see that y must lie on the polar of the point 0lt 02 , 03 with respect to the pitch conic (§ 201) or the locus of all the screws for which Pi = °-