74 THE THEORY OF SCREWS. 



In like manner if ft, 7 be two other screws of the three-system, 



A = xpijrfi -f 



But as 6 belongs to the three-system its co-ordinates must satisfy three 

 linear equations. These we may take to be 



We have thus six linear equations in the co-ordinates of 0. We can therefore 

 eliminate those co-ordinates, thus obtaining a determinantal equation which 

 gives a cubic for x. 



The three roots of this cubic will give accordingly three screws in the 

 three -system which possess the required property. 



Thus we demonstrate that in any three-system there are three principal 

 screws of inertia, and a precisely similar proof for each of the six values of 

 n establishes by induction the important theorem that there are n principal 

 screws of inertia in the screw system of the ?ith order. It is shown in 86 

 that all the roots are real. 



We shall now prove that the Principal Screws of Inertia are co-reciprocal. 

 Let and &amp;lt;/&amp;gt; be two such screws, corresponding to different roots x, x&quot; of 

 the equation in x. 



Then we have 



, , , , , 



pi-x p-2 -m p t x 



Let /* be the screw of the reciprocal system on which the impulsive wrench 

 is generated by the impulse given on 0. 



Then 



, y/*i , y^ , yn* 



&amp;lt;Pl = // , 0-J = - 7, , 06 = - 7, 



PI X p., x p 6 x 



As fi is reciprocal to and X is reciprocal to &amp;lt;f&amp;gt;, we have 



. , 

 / T n 



Pi - % P* X ps-% 



, , 



// T ~77 &quot;T ~r 



pi - as p. 2 -x p 6 - x 



Subtracting these equations and discarding the factor x x&quot;, we get 



__ 

 &quot;\ i i / f\ / 



i- x&quot;) ( ^ 2 - x )( p 2 - x&quot;) ( p 6 - x } ( p s - x&quot;} 



