76 HARMONIC SCREWS. 



T. Hence, T must be a harmonic screw, and as there 

 are only n harmonic screws, it is plain that T must 

 coincide with H n , and that therefore H n is a conjugate 

 screw of inertia, as well as a conjugate screw of the 

 potential, to each one of the remaining n i harmonic 

 screws. Similar reasoning will, of course, apply to 

 each of the harmonic screws taken in succession. 



75. Equations of Motion. We now consider the kine- 

 tical problem, which may be thus stated. A free or 

 constrained rigid body, which is acted upon by a system, 

 of forces, is displaced by an initial twist of small ampli- 

 tude, from a position of equilibrium. The body also 

 receives an initial twisting motion, with a small twist 

 velocity, and is then abandoned to the influence of the 

 forces. It is required to ascertain the nature of its sub- 

 sequent movements. 



Let T represent the kinetic energy of the body, in 

 the position of which the co-ordinates, referred to the 

 principal screws of inertia, are 0,', . . . , n '. Then we 

 have ( 67) : 



/^0/Yl 

 ^ \-dT) _ 



while the potential energy which, as before, we denote 

 by V, is an homogeneous function of the second order 

 of the quantities 0/, . . . . , ft/. 



By the use of Lagrange's method of generalized co- 

 ordinates* we are enabled to write down at once the n 

 equations of motion in the form : 





 dt 



* M<?canique Analytique, vol. i., p. 304. See also Thomson and Tait's 

 Natural Philosophy, vol. i., p. 253. 



