A Treatise on the Theory of Screws

424 THE THEORY OF SCREWS. [388 Let p be the pitch of the screw on which the wrench thus represented lies, and let x, y, z be the co-ordinates of any point on this screw. Then, in the plane of Z the moments of the forces are xY -yX, and if to this be added pX, the whole must equal II. Thus we have the three equations, so well known in statics, F =pX+yZ - zY, G —pY + zX - xZ, H—pZ + xY — yX. The centrifugal acceleration on a point P is, of course, w2PH, where a> is the angular velocity, and PH the perpendicular let fall on the axis. The three components of this force are X', Y', Z', where X' = a? sin 0 (x sin 0 - y cos Ö), Y' = co2 cos 0 (y cos 0 — x sin 0), Z' = a? (z — m sin 2$), and the three moments are F', G', H', where F' = a>2 sin 0 (yz sin 0 + xz cos 0 - 2my cos 0), G' = ®2 cos 0 (- yz sin 0 - xz cos 0 + 2mx sin 0), H'= w2 {(y2 - a?2) sin 0 cos 0 + xy cos 20}. We are now to integrate these expressions over the entire mass, and we employ the following abbreviations (§ 324):__ $xdm = Mx, ■ fydm = My,; jzdm = Mz0; Ry2 ~ ®2) dm = (p2 - p2) M ; fxydm = Mlf; fxzdm=Ml22; jyzdm=Ml2; X = fX'dm; Y= JY'dm; Z = jZ'dm; F = jF'dm; G = jG’dm; H = jH'dm; then, omitting the factor Ma?, we have X= + («0 sin 0 — y, cos 0) sin 0, Y = - (x0 sin 0 - y0 cos 0) cos 0, Z = z„ — m sin 20; ■^ = + sin 0 (l^ sin 0 + l22 cos 0) — 2my0 sin 0 cos 0, G = — cos 0 (Zj2sin 0 + I2cos 0) + 2ma?0 sin 0 cos 0, H = (p,2 - p32) sin 0 cos 0 + r2 cos 20. We can easily verify that FY - GX = 2mXY.