A Treatise on the Theory of Screws

47] SCREW CO-ORDINATES. 41 The direction of the screw of infinite pitch is indicated by the fact that as a twist about it is a translation with components a (a2 — a2), b (a3 — a4), c (a5 — a6), the screw must be parallel to a ray of which these three quantities are proportional to the direction cosines. As the three equations to the screw have disappeared, the situation of the screw is indeterminate. This is of course what might be expected, because a couple is equally efficacious in any position in its plane. 46. A Screw at infinity. If we have «i + a2 = 0; a3 + a4 = 0; a6 + a6 = 0; then the three equations (i), of § (43) will be satisfied for a screw entirely at infinity, no matter what its pitch may be. From this and the last article we see that the three equations »i + a2 = 0; a3 + a4 = 0; a5 + a6 = 0 may mean either a screw of infinite pitch and indefinite position, or a screw of indefinite pitch lying in the plane at infinity. 47. Screws on one axis. The co-ordinates being referred to six canonical co-reciprocals, it is required to determine the co-ordinates of the screws of various pitches which lie on the same axis as a given screw a. We have from § 36 (a + Pa) («i + a2) - dai sin (al) 2a ’ (a +p<a) («1 + a2) - dai sin (ai) 601 “ 2a whence — a>, =^a («i + a.,). We thus have the useful results = (aj + a.); ®2 = a2(«j + aa), ®3 = a3 («3 + «4); = «4 + («3 + «A = a5 (“5 + “«) > = “s +^~2c^ + These formulae may be verified by observing that one of the equations (§ 43) defining co is (co5 + &>6) y - (ws + w4) z = a (wj - w2) - p„ (w, + w2).