DYNAMICS OF A RIGID BODY. 159 



-wrench on a screw r\ belonging to A, then the body will com- 

 mence to twist about the screw 9, of which r\ is the polar with 

 respect to the kinetic complex. 



The screw TJ is, of course, only one of a screw com- 

 plex S of the (7 - n) th order, an impulsive wrench on 

 any one of which would make a body commence to twist 

 about 9 ( 55) ; r\ is, however, the only screw belonging 

 to S which also belongs to A ; a wrench on i\ is the 

 reduced wrench on A, appropriate to a wrench on any 

 other screw belonging to ,5* ( 66). 



1 49 The Potential Complex. If a rigid body which 

 has freedom of the n th order be displaced from a position 

 of stable equilibrium by a twist of given amplitude about 

 & screw 9, of which the co-ordinates referred to the n 

 principal screws of the potential are 1? . . . n , then the 

 potential energy of the new position is proportional to 



Vtff + &C. + V n Z 9 n \ 



If this expression be equated to zero, it denotes a 

 screw complex of the (n - i) th order and second degree, 

 which may be termed the potential complex. 



The potential complex possesses a physical import- 

 ance in every respect analogous to that of the kinetic 

 complex ; by reference to ( 72) the following theorem 

 can be deduced. 



If a rigid body be displaced from a position of equi- 

 librium by a twist about a screw 0, then a wrench acts 

 upon the body in its new position on a screw which is 

 the polar of 9 with respect to the potential complex. 



150 Harmonic Screws. The constructions by which 

 the harmonic screws were determined in the case of the 

 second and the third orders have no analogies in the 

 fourth order. We shall, therefore, here state a general 

 algebraical method by which they can be determined. 



Let /"=o be the kinetic complex, and V=o the 



