A Treatise on the Theory of Screws

13] THE CYLINDROID. 19 the two cases are identical. The amplitudes of the three twists which neutralise are, therefore, proportional to the intensities of the three wrenches which equilibrate. When three twists (or wrenches) neutralise, then a twist (or wrench) equal and opposite to one of them must be the resultant of the other two. once it follows that the laws for the composition of twists and of wrenches must be identical. 13. The Cylindroid. We next proceed to study the composition of twists and wrenches, and we select twists for this purpose, though wrenches would have been equally convenient. A body receives twists about three screws; under what conditions will t e body, after the last twist, resume the same position which it had before the first ? The problem may also be stated thus:—It is required to ascertain the single screw, a twist about which would produce the same effect as any two given twists. We shall first examine a special case, and from it we shall deduce the general solution. -fake, as axes of x and y, two screws a, ß, intersecting at right angles, whose pitches are pa and pß. Let a body receive twists about these screws amplitudes ff cos I and 3' sin I. The translations parallel to the coordinate axes are pa3' cos I and pß3‘ sin I. Hence the axis of the resultant twist makes an angle I with the axis of x; and the two translations may be resolved into two components, of which 3' (pacos21 + pß sin'2/) is parallel to the axis of the resultant twist, while 3' sin I cos I (pa - pß) is perpendicular to the same line. Ihe latter component has the effect of transferring the resultant axis of the rotations to a distance sin I cos I (pa — pß), the axis moving parallel to itself ln a plane perpendicular to that which contains a and ß. The two original twists about a and ß are therefore compounded into a single twist of amplitude 3’ about a screw 3 whose pitch is pa cos21 + pß sin2Z. ihe position of the screw 3 is defined by the equations ( ' , y = x tan I, z = (pa — Pß) sin I cos I. Eliminating I we have the equation z {a? + y2) - {pa - Pß) ®y = o.