A Treatise on the Theory of Screws

125] FREEDOM OF THE SECOND ORDER. 113 If we draw a pencil of four lines through a point parallel to four gene- rators of a cylindroid, the lines forming the pencil will lie in a plane. We may define the anharmonic ratio of four generators on a cylindroid to be the anharmonic ratio of the parallel pencil. We shall now prove the follow- ing theorem :— The anharmonic ratio of four screws on the impulsive cylindroid is equal to the anharmonic ratio of the four corresponding screws on the instantaneous cylindroid. Before commencing the proof we remark that, If an impulsive wrench of intensity F acting on the screw X be capable of producing the unit of twist velocity about A, then an impulsive wrench of intensity Fa> on X will produce a twist velocity a> about A. Let Xlt X2, X:l, X4 be four screws on the impulsive cylindroid, the intensities of the wrenches appropriate to which arc F4a>u F2a>2, .F3w3> Let the four corresponding instantaneous screws be Alt A2, As, A4, and the twist velocities be a>2, <a3) ®.,. Let </>„,, bo the angle on the impulsive cylindroid defining Xm, and let 0,tl. be the angle on the instantaneous cylindroid defining Am. If three impulsive wrenches equilibrate, the corresponding twist velocities neutralize by the second law of motion : hence (§ 14) certain values of «i, w2> &>3, must, satisfy the following equations:— _ (l>2 _______&>3 sin (0., — 03) sin (03 — #i) sin (0j — 02) -A'jWj F.,<O2 __ F__________ sin (</>2 — </>3) sin (</>3 — fa) sin (</>( — </>2) &>2 _ &>3 ________">l sin (/( (()- sinOV- ^) ~ sin Ök “ ’ F2(o2 __ F3ms _ FttOf sm(fa - fa) ~ sin (fa - fa) sin (fa -fa)’ whence sin (04 — 0..) sin (0-. — 00 _ sin (fa — fa) sin (fa — fa) sm(0f- 0i) sin(0i - 02) ~ sin (fa3 - fa) sin (fa - fa) ’ which proves the theorem. If we are given three screws on the impulsive cylindroid, and the corresponding three screws on the instantaneous cylindroid, the connexion between every other corresponding pair is, therefore, geometrically deter- mined. B. 8