5 2 THE PRINCIPAL SCREWS OF INERTIA. 



twisting about S this screw must belong to the complex 

 specified by A Jy &c. A n . The n impulsive wrenches on 

 AI, &c. A n will produce twisting motions about the same 

 screws, but these twisting motions are to compound into 

 a twisting motion on S. It follows that the component 

 twist velocities Si, &c. S n ' must be proportional to the 

 intensities Si", &c. S n ". But if this were the case, then 

 every screw of the complex would be a principal screw 

 of inertia ; for let X be any impulsive screw, and suppose 

 that Y is the corresponding instantaneous screw, the 

 components of JTon AI, &c. A n , have intensities JT/ 7 , &c. 

 X n ", these will generate twist velocities equal to 



o / c / 



X " &c X " 



g // i > ^ S ' 



and these quantities must equal the components of the 

 twist velocity about Y. But the ratios 



are all equal, and hence the twist velocities of the com- 

 ponents on the screws of reference of the twisting motion 

 about Fmust be proportional to the intensities of the 

 components on the same screws of reference of the 

 wrench on X. Remembering that twisting motions and 

 wrenches are compounded by the same rules, it follows 

 that Y and X must be identical. 



As it is not generally true that all the screws of the 

 complex defining the freedom possess the property 

 enjoyed by a principal screw of inertia, it follows that the 

 number of principal screws of inertia must be generally 

 equal to the order of the freedom. 



58. Kinetic Energy. The twisting motion of a rigid 

 body with freedom of the n th order may be completely 

 specified by the twist velocities of the components of the 



