HARMONIC SCREWS. 8 1 



is an harmonic screw, and hence we have the follow- 

 ing theorem : 



If a rigid body occupy a position of stable equili- 

 brium under the influence of a system of forces, as 

 restricted in 6, then n harmonic screws can be selected 

 from the screw complex of the n th order, which defines 

 the freedom of the body, and if the body be displaced 

 from its position of equilibrium by a twist about a har- 

 monic screw, and if it also receive a small initial twist 

 velocity about the same screw, then the body will continue 

 for ever to perform twist oscillations about that harmonic 

 screw, and the amplitude of the twist will be always 

 equal to the arc of a certain circular pendulum, which has 

 an appropriate length, and was appropriately started. 



The integrals in their general form prove the follow- 

 ing theorem : 



A rigid body is slightly displaced by a twist from 

 a position of stable equilibrium under the influence of a 

 system of forces, and the body receives a small initial 

 twisting motion. The twist, and the twisting motion, 

 may each be resolved into their components on the 

 n harmonic screws : n circular pendulums are to be con- 

 structed, each of which is isochronous with one of the 

 harmonic screws. All these pendulums are to be started 

 at the same instant as the rigid body, each with an arc, 

 and an angular velocity equal to the initial amplitude of 

 the twist, and the twist velocity, which has been assigned 

 to the corresponding harmonic screw, as its share of the 

 initial circumstances. To ascertain where the body 

 would be at any future epoch, it will only be necessary 

 to calculate the arcs of the n pendulums for that epoch, 

 and then give the body twists from its position of equili- 

 brium about the harmonic screws, whose amplitudes are 

 equal to these arcs. 



The reader will observe that the solution to which 



G 



