CHAPTER II. 

 THE CYLINDROID. 



12. Introduction. We shall now ascertain the laws 

 according to which twists (and wrenches) must be com- 

 pounded together, that is to say, we shall determine 

 the single screw, one twist (or wrench) about which will 

 produce the same effect on the body as two or more 

 given twists (or wrenches) about two or more given 

 screws. It will be found to be a fundamental point of the 

 present theory that the rules for the composition of twists 

 and of wrenches are identical.* 



13. On the Virtual Coefficient of a Pair of Screws. 

 Suppose a rigid body be acted upon by a wrench 

 on a screw )3, of which the intensity is /3". Let the 

 body receive a twist of small amplitude a' around a 

 screw a. It is proposed to find an expression either for 

 the energy required to effect the displacement, or the 

 work done if the displacement be permitted. 



Let d be the shortest distance between a and j3, and 

 let O be the anglef between a and )3. Take a as the axis 

 of x y the common perpendicular to a and /3 as the axis 



* That the source of the analogy between the composition of forces and of 

 rotations lies in the general principles of virtual velocities, has been proved by 

 Rodrigues (Liouville's Journal, t. 5, 1840, p. 436). 



f Perhaps the best convention to distinguish between O and its supplement 

 is the following : Suppose the common perpendicular to be an ordinary right- 

 handed screw, and that there is a nut on this screw to which a is attached. 

 If, then, the nut be turned so as to make a approach ft (that is, to make the 

 length of the common perpendicular diminish), the angle less than TT through 

 which a has turned when it has become parallel to ft is the angle O. 



