106] HARMONIC SCREWS. 99 



106. Discussion of the Results. 



For the position of the body before its displacement to have been one of 

 stable equilibrium, it is manifest that the co-ordinates must not increase 

 indefinitely with the time, and therefore all the values of s 2 must be essen 

 tially positive, since otherwise the values of #/, .. n would contain expo 

 nential terms. 



The 2n arbitrary constants are to be determined by the initial circum 

 stances. The initial displacement is to be resolved into n twists about the n 

 screws of reference ( 95). This will provide n equations, by making t = 0, 

 and substituting for Oi,...0 n , in the equations just mentioned, the amplitudes 

 of the initial twists. The initial twisting motion is also to be resolved into 

 twisting motions about the n screws of reference. The twist velocities of 



JQ I JQ &amp;gt; 



these components will be the values of ; , . . . , ~, , when t = ; whence 



we have n more equations to complete the determination of the arbitrary 

 constants. 



If the initial circumstances be such that the constants H.,, ..., H n are all 

 zero, then the equations assume a simple form : 



0i =f u #! sin (s^ + c), 



O n =/!?! sin (sj + c). 



The interpretation of this result is very remarkable. We see that the 

 co-ordinates of the body are always proportional to f u , ...,f ln , hence the 

 body can always be brought from the initial position to the position at any 

 time by twisting it about that screw, whose co-ordinates are proportional to 

 fu&amp;gt; &amp;gt;f-m\ but, as we have already pointed out, the screw thus defined is 

 a harmonic screw, and hence we have the following theorem : 



If a rigid body occupy a position of stable equilibrium under the 

 influence of a conservative system of forces, then n harmonic screws can be 

 selected from the screw system of the nth order, which defines the freedom 

 of the body, and if the body be displaced from its position of equilibrium 

 by a twist about a harmonic screw, and if it also receive any small initial 

 twist velocity about the same screw, then the body will continue 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 following 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 



7-2 



