1 1 2 DYNAMICS OF A RIGID BODY. 



to consider the distribution of the parameter v a upon 

 the screws of a cylindroid. It appears from ( 72) 

 that if v ly v- 2 denote the values of the quantity v a for 

 each of two conjugate screws of the potential, and if 

 01, a 2 denote the intensities of the components on the two 

 conjugate screws of a wrench of unit intensity on a screw 

 a, which lies upon the cylindroid, that then 



From the centre of the cylindroid draw two straight 

 lines parallel to the pair of conjugate screws of the 

 potential, and with these lines as axes of x and y con- 

 struct the ellipse, of which the equation is 



where H is any constant. If r be the radius vector in 

 this ellipse, we have 



x j y 



= cti and = as ; 



whence by substitution we deduce 



2 _# 



which proves the following theorem : 



The linear parameter v a on any screw of the cylin- 

 droid is inversely proportional to the parallel diameter 

 of a certain ellipse, and a pair of conjugate screws of 

 the potential are parallel to a pair of conjugate diameters 

 of the same ellipse. 



This ellipse maybe called the ellipse of the potential. 



The major and minor axes of the ellipse of the poten- 

 tial are parallel to screws upon the cylindroid, which, for 

 a twist of given amplitude, correspond to a maximum 

 and minimum potential energy. 



When the body is slightly displaced from its posi- 

 tion of equilibrium by the action of a wrench of given 

 small intensity on a given screw 17, the twist which 

 the body executes in assuming its new position is per- 



