200 THE THEORY OF SCREWS. [200- 



movcs over the cylindroid and the corresponding point moves over the 

 straight line. Hence we obtain the following important result : 



The several screws on a cylindroid correspond to the points on a straight 

 line. 



In general two cylindroids have no screw in common. If, however, the two 

 cylindroids be each composed of screws taken from the same three-system, 

 then they will have one screw in common. This is demonstrated by the 

 fact that the two straight lines corresponding to these cylindroids necessarily 

 intersect in a point which corresponds to the screw common to the two 

 surfaces. 



Three twist velocities about three screws will neutralize and produce 

 rest, provided that the three corresponding points lie in a straight line, and 

 that the amount of each twist velocity is proportional to the sine of the 

 angle between the two non-corresponding screws. 



Three wrenches will equilibrate when the three points corresponding to 

 the screws are collinear, and when the intensity of each wrench is propor 

 tional to the sine of the angle between the two non -corresponding screws. 



201. The Screws of the Three-system. 



In any three-system there are three principal screws at right angles to 

 each other, and intersecting in a point ( 173). It is natural to choose these 

 as the screws of reference, and also as the axes for Cartesian co-ordinates. 

 The pitches of these screws are p 1} p.,, p 3 , and we shall, as usual, denote the 

 screw co-ordinates by 0,, a , 3 . The displacement denoted by this triad of 

 co-ordinates is obtained by rotating the body through angles 1} 0. 2 , 3 around 

 three axes, und then by translating it through distances p^, p,0. 2 , p 3 3 parallel 

 to these axes. As these quantities are all small, we have, for the displace 

 ments produced in a point x, y, z, 



Sy = p. 2 2 + x0 3 - z6 l , 

 8z = p s 3 + y0 1 -x0 2 ; 



these displacements correspond to a twist about a screw of which the axis 

 has the equations 



pA + z0y - y0 3 _ p 2 2 + x0 3 - z0 l p 3 3 + yfl - x0. 2 



0i 0, ~0T~ 



while the pitch p is thus given : 



