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. 

 Hence 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 

 the 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. 



Take, as axes of x and y, two screws a, /3, intersecting at right angles, 

 whose pitches are p a and pp. Let a body receive twists about these screws 

 of amplitudes & cos I and 6 sin I. The translations parallel to the coordinate 

 axes are p a 6 cos I and p$& 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 (p a cos 2 1 + pp sin 2 /) is parallel to the axis of the 

 resultant twist, while 6 sin I cos I (p a - pp) is perpendicular to the same line. 

 The latter component has the effect of transferring the resultant axis of the 

 rotations to a distance sin I cos I (p a pp), the axis moving parallel to itself 

 in a plane perpendicular to that which contains a and /3. The two original 

 twists about a and /3 are therefore compounded into a single twist of 

 amplitude & about a screw 6 whose pitch is 



The position of the screw 6 is defined by the equations 



y x tan /, 

 z = (p a pp) sin I cos I. 



Eliminating I we have the equation 



22 



