54] THE REPRESENTATION OF THE CYLINDROID BY A CIRCLE. 49 



The circular representation of the cylindroid is now complete. We see 

 how the pitch of each screw is given, and how the perpendicular distance 



and the angle between every pair of screws can be concisely represented. 

 We may therefore study the dynamical and kinematical properties of the 

 cylindroid by its representative circle. We commence by proving a funda 

 mental principle very analogous to an elementary theorem in Statics. 



54. The Triangle of Twists. 



It has been already shown ( 14) that any three screws on the cylindroid 

 possess the following property : 



If a body receive twists about three screws, so that the amplitude of 

 each twist is proportional to the sine of the angle between the two non- 

 corresponding screws, the body, after the last twist, will be restored to where 

 it was before the first. 



With the circular representation of the cylindroid we transform this 

 theorem into the following: 



If any three screws, A, B, C (Fig. 8), be taken on the circle, and if twists 

 be applied to a body in succession, so that the amplitude of each twist is 

 proportional to the opposite side of the triangle ABC, then the body will 

 be restored by the last twist to the place it had before the first. 



From the analogy of wrenches, and of twist velocities to twists, we are 

 also able to enunciate the following theorems : 



If wrenches upon the three screws A, B, C be applied to any rigid body, 

 then these wrenches will equilibrate, provided that the intensity of each is 

 proportional to the opposite side of the triangle. 



