A Treatise on the Theory of Screws

22 THE THEORY OF SCREWS. [14- These four quantities vanish if O' </> y/f' sin (m — ri) sin (w — I) sin (7 — m) ’ and hence the fundamental property of the cylindroid has been proved. The cylindroid affords the means of compounding two twists (or two wrenches) by a rule as simple as that which the parallelogram of force pro- vides for the composition of two intersecting forces. Draw the cylindroid which contains the two screws; select the screw on the cylindroid which makes angles with the given screws whose sines are in the inverse ratio of the amplitudes of the twists (or the intensities of the wrenches); a twist (or wrench) about the screw so determined is the required resultant. The amplitude of the resultant twist (or the intensity of the resultant wrench) is proportional to the diagonal of a parallelogram of which the two sides are parallel to the given screws, and of lengths proportional to the given ampli- tudes (or intensities). 15. Particular Cases. If pa =pp the cylindroid reduces to a plane, and the pitches of all the screws are equal. If all the pitches be zero, then the general property of the cylindroid reduces to the well-known construction for the resultant of two intersecting forces, or of rotations about two intersecting axes. If all the pitches be infinite, the general property reduces to the construction for the composition of two translations or of two couples. 16. Cylindroid with one Screw of Infinite pitch. Let OP, Fig. 2, be a screw of pitch p about which a body receives a small twist of amplitude w. p Fig. 2 . Let OR be the direction in which all points of the rigid body are trans- lated through equal distances p by a twist about a screw of infinite pitch