128] FREEDOM OF THE SECOND ORDER. 115 



the screw a. with the unit of twist velocity, the kinetic energy is found by 

 multiplying the mass of the body into the square of the line u a . 



We are now going to consider the distribution of this magnitude u a on 

 the screws of a cylindroid. If we denote by u lt u^ the values of ti a for any 

 pair of conjugate screws of inertia on the cylindroid (81), and if by a l , a, 

 we denote the intensities of the components on the two conjugate screws of 

 a wrench of unit intensity on a, we have ( 97) 



From the centre of the cylindroid draw two straight lines parallel to the 

 pair of conjugate screws of inertia, and with these lines as axes of as and y 

 construct the ellipse of which the equation is 



U-fX* + M 2 2 7/ 2 = H, 



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

 ( 35) 



x y 



- = ! and - = a,, ; 



r r 



whence by substitution we deduce 



which proves the following theorem: 



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

 proportional to the parallel diameter of a certain ellipse, and a pair of 

 conjugate screws of inertia on the cylindroid are parallel to a pair of 

 conjugate diameters of the same ellipse. This ellipse may be called the 

 ellipse of inertia. 



The major and minor axes of the ellipse of inertia are parallel to screws 

 upon the cylindroid, which for a given twist velocity correspond respectively to 

 a maximum and minimum kinetic energy. 



An impulsive wrench on a screw 77 acts upon a quiescent rigid body 

 which has freedom of the second order. It is required to determine the 

 screw 6 on the cylindroid expressing the freedom about which the body 

 will commence to twist. 



The ellipse of inertia enables us to solve this problem with great facility. 

 Determine that one screw &amp;lt;f&amp;gt; on the cylindroid which is reciprocal to ?? ( 26). 

 Draw a diameter D of the ellipse of inertia parallel to &amp;lt;. Then the required 

 screw 6 is simply that screw on the cylindroid which is parallel to the 

 diameter conjugate to D in the ellipse of inertia. 



The converse problem, viz., to determine the screw 77, an impulsive wrench 



82 



