129] FREEDOM OF THE SECOND ORDER. 117 



wrench of unit intensity on a screw a, which also lies upon the cylindroid, 

 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 construct the ellipse, of which the equation is 



vi*a? + v./y* = H, 

 where H is any constant. If r be the radius vector in this ellipse, we have 



x y 



= a, and - = a., ; 

 r r 



whence by substitution we deduce 



-- i ^;* a :; 



which proves the following theorem : 



The linear parameter v a on any screw of the cylindroid is inversely pro 

 portional 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 may be called the ellipse of the potential. 



The major and minor axes of the ellipse of the potential 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 has to relinquish its original position of equilibrium by 

 the addition of a wrench on a screw 77 to the forces previously in operation, 

 the twist by which the body may proceed to its new position of equilibrium 

 is about a screw 0, which can be constructed by the ellipse of the potential. 

 Determine the screw &amp;lt;/&amp;gt; (on the cylindroid of freedom) which is reciprocal to 

 77 ( 26), then &amp;lt;/&amp;gt;, and the required screw 0, are parallel to a pair of conjugate 

 diameters of the ellipse of the potential. 



The common conjugate diameters of the pitch conic, and the ellipse of 

 the potential, are parallel to the two screws on the cylindroid, which we 

 have designated the principal screws of the potential ( 101). 



When a body is disturbed from its position of equilibrium by a small 

 wrench upon a principal screw of the potential, then the body could be moved 

 to the new position of equilibrium required in its altered circumstances by a 

 small twist about the same screw. 



