DYNAMICS OF A RIGID BODY. 109 



We have thus arrived by a special process at the two 

 principal screws of inertia posssesed by a body which 

 has freedom of the second order. This is, of course, 

 a particular case of the general theorem of 5 1 . We 

 shall show in the next section how these screws can 

 be determined in another manner. 



102. The Ellipse of Inertia. We have seen ( 59) 

 that a linear parameter u^ may be conceived appropriate 

 to each screw a of a complex, so that when the body is 

 twisting about 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^ 



We are now going to consider the distribution of this 

 magnitude on u a the screws of a cylindroid. If we denote 

 by Ui, u<i the values of u a for any pair of conjugate screws 

 of inertia on the cylindroid ( 54), and if by <n, a 2 we 

 denote the intensities of the components on the two con- 

 jugate screws of a wrench of unit intensity on a, we have 



From the centre of the cylindroid draw two lines 

 parallel to the pair of conjugate screws of inertia, and 

 with these lines as axes of x and y construct the ellipse 

 of which the equation is 



u\X" 4" u<^y = j. , 



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

 this ellipse, we have 



y /y 



= cti and = a 2 ; 



r r 



whence by substitution we deduce 



2 _H 



which proves the following theorem : 



