8o HARMONIC SCREWS. 



76. 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 there- 

 fore all the values of s z must be essentially positive, since 

 otherwise the values of 0'j, . . . 0' would contain expo- 

 nential terms. 



The 2n arbitrary constants are to be determined by 

 the initial circumstances. The initial displacement is 

 to be resolved into n twists about the n screws of refer- 

 ence ( 65). This will provide n equations, by making 

 t = o, and substituting for 0/, #', in the equations just 

 mentioned, the amplitudes of the initial twists. The 

 initial twisting motion is also to be .resolved into twist- 

 ing motions about the n screws of reference. The twist 

 velocities of these components will be the values of 



jf\ r jf\ / 



aui - ctu n 1,1 i 



, &c., =7-, when / = o ; whence we have n more 

 at at 



equations to complete the determination of the arbitrary 

 constants. 



If the initial circumstances be such that the con- 

 stants JJ 2 , > H n are all zero, then the equations as- 

 sume a very simple form .: 



ft' /!!/?! -sin (r i/+i), 



&c. 

 O n ' 



The interpretation of this result is very remarkable. 

 We see that the co-ordinates of the body are always 

 proportional to f uy . . . . ,/i n ; hence the body can al- 

 ways be brought from the initial position to the position 

 at any time by twisting it about that screw, whose 

 co-ordinates are proportional to f lly . . . .,/!; but, as 

 we have already pointed out, the screw thus defined 



