DYNAMICS OF A RIGID BODY. 107 



on the impulsive cylindroid defining X my and let 6 m be 

 the angle on the instantaneous cylindroid defining A m . 



If three impulsive wrenches equilibrate, the corres- 

 ponding twist velocities neutralise : hence ( 1 7) it must be 

 possible for certain values of on, o> 2 , w 3 , 014 to satisfy the 

 following equations : 



(jt}\ fiJ*> &^3 



sin (ft - ft) sin (03 - 00 sin (ft - ft)' 



whence 



sin (0j - ft) sin (ft - ft) = sin (fa -fa) sin (fa -fa] 

 sin (ft - ft) sin (0 4 - ft) ~ sin (0 3 - 00 sin (0 4 - 2 ) J 



which proves the theorem. 



If, therefore, we are given three screws on the impul- 

 sive cylindroid, and the corresponding three screws on 

 the instantaneous cylindroid, the connexion between 

 every other corresponding pair is geometrically] deter- 

 mined. 



100. Reaction of Constraints. Whatever the con- 

 straints may be, their reaction produces an impulsive 

 wrench ^ upon the body at the moment when the 

 impulsive wrench X l acts. The two wrenches X and 

 R compound into a third wrench Y\. If the body were 

 free, Y l is the impulsive wrench to which the instanta- 

 neous screw A l would correspond. Since X it X^ X z 

 are cocylindroidal, A l9 A 2 , A 3 must be cocylindroidal, 



