192 THE THEORY OF SCREWS. [193- 



193. Quadric of the Potential. 



A body which has freedom of the third order is in equilibrium under 

 the influence of a conservative system of forces. The body receives a twist 

 of small amplitude 9 about a screw 9 of the screw system. It is required 

 to determine a geometrical representation for the quantity of work which 

 has been done in effecting the displacement. We have seen that to each 

 screw 9 corresponds a certain linear parameter V B ( 102), and that the work 

 done is represented by 



F^&\ 



We have also seen that the quantity v e ~ may be represented by 



where l} 2 , 9 3 are the co-ordinates of the screw 9 referred to three conjugate 

 screws of the potential, and v lt v. 2 , v s , denote the values of v g for each of the 

 three screws of reference ( 102). 



Drawing through the centre of the pitch quadric three axes parallel to 

 the three screws of reference, we can then construct the quadric of which 

 the equation is 



v^x z + v?y- + v 3 2 z- = H, 



which proves the following theorem : 



The work done in giving the body a twist of given amplitude from a 

 position of equilibrium about any screw of a system of the third order, is 

 proportional to the inverse square of the parallel diameter of a certain 

 quadric which we may call the quadric of the potential, and three conjugate 

 diameters of this quadric are parallel to three conjugate screws of the 

 potential in the screw system. 



194. The Principal Screws of the Potential. 



The three common conjugate diameters of the pitch hyperboloid, and 

 the quadric of the potential, are parallel to three screws of the system 

 which we call the principal screws of the potential. If the body be 

 displaced by a twist about a principal screw of the potential from a 

 position of stable equilibrium, then the reduced wrench which is evoked 

 is upon the same screw. 



The three principal screws of the potential must not be confounded with 

 the three screws of the system which are parallel to the principal axes of 

 the ellipsoid of the potential. The latter are the screws on which a twist 

 of given amplitude requires a maximum or minimum consumption of 

 energy, and they are rectangular, which, of course, is not in general the 

 case with the principal screws of the potential. 



