220 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. 

 From 32) with a 2 a = y, 



Since the cylindroid is thus determined by 31) and 33), a twist 

 about PI can be resolved into a twist about p x and one about p y . 

 A twist about p 2 can be likewise resolved. The two components 

 about p x add together, so do those about p y , and since the resultant 

 of any twists about p x and p y lies on the cylindroid , the resultant 

 of p 1 and p 2 does. Its direction can be found, since the amplitudes E 

 of the two twists about p^p 2 compound by the parallelogram law, 

 hence the angle made by the resultant with the axes is known. The 

 pitch is then found from the pitch -conic. 



65. Work of Wrench in Producing a Twist. Let us find 

 an expression for the work done during a twist of amplitude R k 

 about a screw of pitch p k by a wrench of intensity Ef about another 

 screw of pitch p f . We already know the work done by a force in 

 a translation, namely, it is equal to the product of the magnitudes 

 by the cosine of the included angle. If the force is R f and the 

 translation (rotation -couple) is $#, we have 



W = E f S k cos(E f S k ). 



Notice that the vector of one system is multiplied by the vector- 

 couple in the other. 



We can find the work done by the force -couple in a rotation 

 about its axis. Apply the couple so that one of its members P 

 passes through the axis of rotation. In a rotation this member does 

 no work, for its point of application is at rest, while that of the 

 other member Q moves in a rotation a distance da, where d is the 

 arm of the couple. Accordingly the work is W=Pdc3 which is 

 equal to the product of the twist by the moment of the couple. 

 Here again we multiply the vector of one system by the vector- 

 couple of the other. 



If the axis of rotation is perpendicular to the axis of the couple, 

 the motion is perpendicular to the force, and no work is done. Hence 

 we must take the resolved part of the couple on the vector, as before. 



We can now find the work of a wrench during a twist. The 

 work of the force in the displacement S k is jR/^cosa, a being the 



angle between the two screws. The work of the couple Sf = -^ E f 

 in the rotation E k is 



Pf 

 S f E k cos a = ~ E f E k cos cc. 



