242 THE THEORY OF SCREWS. [227, 



belonging to A, then the body will commence to twist about the screw 6, of 

 which V) is the polar with respect to the quadratic n-system composed of 

 the imaginary screws about which the body would twist with zero kinetic 

 energy. 



If a rigid body which has freedom of the nth order be displaced from a 

 position of stable equilibrium under the action of a system of forces by a 

 twist of given amplitude about a screw 6, of which the co-ordinates referred 

 to the n principal screws of the potential are O l ,... n , then the potential 

 energy of the new position may, as we have seen ( 103) be expressed by 



If this expression be equated to zero, it denotes a quadratic rc-system, 

 which is of course imaginary. We may term it the potential quadratic 



w-system. 



The potential quadratic w-system possesses a physical importance in 

 every respect analogous to that of the kinetic quadratic n-system : by 

 reference to ( 102) the following theorem can be deduced. 



If a rigid body be displaced from a position of stable equilibrium by a twist 

 about a screw 6, then a wrench acts upon the body in its new position on 

 a screw which is the polar of 6 with respect to the potential quadratic 

 w-system. 



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 U=0 be the kinetic quadratic ?i-system, and F=0 the potential 

 quadratic w-system, then it follows from a well-known algebraical theorem 

 that one set of screws of reference can in general be found which will reduce 

 both U and V to the sum of n squares. These screws of reference are the 

 harmonic screws. 



We may here also make the remark, that any quadratic w-system can 

 generally be transformed in one way to the sum of n square terms with 

 co-reciprocal screws of reference; for if U and p e be transformed so 

 that each consists of the sum of n square terms, then the form for the 

 expression of p e ( 38) shows that the screws are co-reciprocal. 



228. On the degrees of certain surfaces. 



We have already had occasion ( 210) to demonstrate that the general 

 condition that two screws shall intersect involves the co-ordinates of each 



