THE PRINCIPAL SCREWS OF INERTIA. 79 



on A 1 will make the body commence to twist about 4,. Hence A l and also 

 A. 2 and A 3 are principal screws of inertia. 



We shall now show that with the exception of the ?? screws here deter 

 mined, generally no other screw possesses the property. Suppose another 

 screw S were to possess this property. Decompose the wrench on S into n 

 wrenches of intensities S&quot;, ... S n &quot; on A lt ... A n ; this must be possible, 

 because if the body is to be capable of twisting about S this screw must 

 belong to the system specified by A lt ... A n . The n impulsive wrenches on 

 A lt ... 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 $ ... &amp;lt;S n must be proportional to the 

 intensities $/ , ... S n &quot;. But if this were the case, then every screw of the 

 system would be a principal screw of inertia ; for let X be any impulsive 

 screw of the system, and suppose that Y is the corresponding instantaneous 

 screw, the components of X on A lt ... A nt have intensities X&quot;, ... X n &quot;, 

 these will generate twist velocities equal to 



c* C 



Q! y n *-*M -y II 



-~-jj A. i , . . . // -A- n &amp;gt; 

 Of 



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 components on the screws 

 of reference of the twisting motion about Y must 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 system 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. 



88. Kinetic Energy. 



The twisting motion of a rigid body with freedom of the nth order may 

 be completely specified by the twist velocities of the components of the 

 twisting motion on any n screws of the system defining the freedom. If the 

 screws of reference be a set of conjugate screws of inertia, the expression for 

 the kinetic energy of the body consists of n square terms. This will now be 

 proved. 



If a free or constrained rigid body be at rest in a position L, and if the 



