140 DYNAMICS OF A RIGID BODY. 



where ft, ft, ft are the co-ordinates of the screw referred 

 to three conjugate screws of the potential, and v ly v 2y v 3 , 

 denote the values of ve for each of the three screws of 

 reference ( 72). 



Drawing through the centre of the pitch quadric three 

 axes parallel to the three screws of reference, we can 

 then construct the ellipsoid of which the equation is 



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 complex of the third order, is proportional to 

 the inverse square of the parallel diameter of a certain 

 ellipsoid which we may call the ellipsoid of the potential, 

 and three conjugate diameters of this ellipsoid are paral- 

 lel to three conjugate screws of the potential in the screw 

 complex. 



127. The Principal Screws of the Potential. The three 

 common conjugate diameters of the pitch hyperboloid, 

 and the ellipsoid of the potential, are parallel to three 

 screws of the complex which are what we call the prin- 

 cipal 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 complex 

 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 mini- 

 mum consumption of energy, and they are rectangular, 

 which, of course, is not in general the case with the 

 principal screws of the potential. 



128. Wrench evoked by Displacement. By the aid of 



