A Treatise on the Theory of Screws

386] THE THEORY OF PERMANENT SCREWS. 421 Let us take any two screws on the cylindroid a and ß, and let their co- ordinates, when referred to the absolute screws of inertia, be ai, ... a„, and ßx, ... ßt. Then any other screw on the cylindroid, about which the body has been displaced by a twist, by components 0J on a and 02' on ß will have, for co- ordinates, aßi + ßA', • • • «A' + ßA', and the screw about which the body is twisting, with a twist velocity d, will have, for co-ordinates, CtjØj + ßÄ, ... otÅ + ßÄ- It readily appears that, so far as the terms involving 0^ and 02 are concerned, the kinetic energy is the expression (ø1'é2-ø.'ø1)(Aé1 + Bfo, where A = + be (a, + a2) [(a5 - a6) (ß3 - Ä) - (a3 - a4) (ß6 - &)] + (62 - c2) (a3 + a4) (a5 + a6) (ft + ß2) + ca (a3 + a4) [(«i — «2) (ßt> ~ ße) ~ («» ~ “<>) (& ~ &)] + (c2 - a2) (as + a6) (ax + a2) (ß3 + ß4) + ab (a5 + a6) [(a3 - a4) (ßi - ß3) ~ (ai ~ “s) (& - &)] + (a2 - b2) (ctj + a2) (a3 + a4) (/?s + /36), B = + be (ß, + ß>) [(<*» - a6) (Ä> - ft) - («3 - «4) (ß, - &)] — (b2 - c2) (ax + a3) (ß3 + ßi) (ßs + ßs) + ca (ß3 + ßß [(«i - a2) (ßs - ßs) - (a6 - «e) (ßi ~ ßi)] - (c2 - a2) (a3 + a4) (ß, + ßs) (ßi + ß?) + ab (ßs + ß6) [(a3 - a4) (Ä - ß>) - («! - «2) (Ä - Ä)] — (a2 — b2) (a5 + a6) + ß2) (ß3 + ßi). 386. The Permanent Screw. We now write the equations of motion for a body which has two degrees of freedom, and is unacted upon by any force, the screws of lefeience being the two principal screws of inertia. We have, from the general equations (§ 367) = d^,,