10] THE CYLINDROID. 17 



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*. 



10. The Virtual Coefficient. 



Suppose a rigid body be acted upon by a wrench on a screw ft, of which the 

 intensity is ft&quot;. Let the body receive a twist of small amplitude a around a 

 screw a. It is proposed to find an expression for the energy required to effect 

 the displacement. 



Let d be the shortest distance between a and ft, and let 6 be the angle 

 between a and ft. Take a as the axis of x, the common perpendicular to a. 

 and ft as the axis of z, and a line perpendicular to x and z for y. If we 

 resolve the wrench on ft into forces X, Y. Z, parallel to the axes, and couples 

 of moments L, M, N, in planes perpendicular to the axes we shall have 



X=ft&quot;cosO; Y=ft&quot;smO; Z=0; 

 L = p pfi cos - ft&quot;d sin ; M = ft&quot; Pii sin + ft&quot;d cos ; 



N = 0. 



We thus replace the given wrench by four wrenches, viz., two forces and 

 two couples, and we replace the given twist by two twists, viz., one rotation 

 and one translation. The work done by the given twist against the given 

 wrench must equal the sum of the eight quantities of work done by each of 

 the two component twists against each of the four component wrenches. 

 Six of these quantities are zero. In fact a rotation through the angle a 

 around the axis of x can do work only against L, the amount being 

 aft&quot; (p ft cos - d sin 0). 



The translation p a a parallel to the axis of x can do work only against 

 X, the amount being 



a ft&quot;p a cos 0. 



Thus the total quantity of work done is 



aft&quot; {(pa+pp) cos d sin 0}. 

 The expression 



i [(P + PP) cos d sin 0] 

 is of great importance in the present theoryf. It is called the virtual 



* That the analogy between the composition of forces and of rotations can be deduced from 

 the general principle of virtual velocities has been proved by Rodrigues (Liouville s Journal, t. 5, 

 1840, p. 436). 



+ The theory of screws has many points of connexion with certain geometrical researches on the 

 linear complex, by Pliicker and Klein. Thus the latter has shown (Mathematische Annalen, Band 

 n., p. 368 (1869)), that if p &nd p s be each the &quot;Hauptparameter&quot; of a linear complex, and if 



(P a +Pft) cos O-d sin = 0, 



where d and relate to the principal axes of the complexes, then the two complexes possess a 

 special relation and are said to be in &quot; involution.&quot; 



B. 2 



