221] FREEDOM OF THE FOURTH ORDER. 231 



The co-ordinates of are proportional to 



/// , rt/rr r\ . ill , 5/&quot;S&amp;gt; 



a a x + p & + 7 71 + o di 



/// . nni rt i i 



a a + /3 #, + 7 7e + &amp;lt;% 



As # is to be a principal screw of inertia it follows that the expressions 

 last written multiplied severally by p l} ...,p 6 must be proportional to the 

 intensities of the impulsive wrenches received by the body : whence we have 

 the following equations in which h is a quantity which is the same for each 

 of the co-ordinates. 



i + 7&quot; 7i + 8 &quot;^ = *&quot; i + P&quot;& + 7&quot;Vi + S/// 



We are now to multiply these equations by a u . .., a fi respectively, and 

 add. If we repeat the process using p\, ...,/3 6 ; 71, ..., 7el !,..., S 6 ; 

 X ls ...,X 6 ; fr, ...,fj, 6 and if we remember that a is reciprocal to /3, 7, 8 

 because the system is co-reciprocal and that a is reciprocal to A, and p, 

 because X and //, belong to the reciprocal system, then observing that like 

 conditions hold for ft, 7, and S, we have the equations 



// S7 1 X 1 +/ // S7,/* 1 =0, 

 / S8 1 X 1 + /i // SV 1 =0, 

 V +/ // S\ 1 ^ 1 =0, 



From these equations a &quot;, /S&quot; , 7 &quot;, 8 &quot;, X&quot; , /*&quot; can be eliminated and the 

 result is to give a biquadratic for h. Thus we have the four roots for the 

 equation. Each of these roots will give a corresponding set of values for 

 &quot; , &quot;, 7 &quot;, 8 &quot;, X &quot;, p&quot; thus we obtain 



which are proportional to the co-ordinates of the corresponding principal 

 screw of inertia. 



The values of X &quot; and ///&quot; determine the impulsive reaction of the con 

 straints. 



221. Application of Euler s Theorem. 



It may be of interest to show how the co-ordinates of the instantaneous 



