235] FREEDOM OF THE FIFTH ORDER. 253 



We can easily verify as in 84 that these five screws are co-reciprocal and 

 are also conjugate screws of inertia. 



It is assumed in the deduction of this quintic that all the quantities 

 XJ...XB are different from zero. If one of the quantities, suppose X l5 had 

 been zero this means that the first absolute principal Screw of Inertia 

 would belong to the w-system expressing the freedom. 



Let us suppose that Xj = then the equations are 



Of course one solution of this system will be V&quot; = 0, , = ... = 0. This 

 means that the first absolute principal Screw of Inertia is also one of the 

 principal Screws of Inertia in the rc-system, as should obviously be the case. 

 For the others = and we have an equation of the fourth degree in 



, 



P* - % 



In the general case we can show that there are no imaginary roots in the 

 quintic, for since the screws 



^1 ^-2 X 



and 



p 1 - x p 2 - x p s - x&quot; 

 are conjugate screws of inertia, we must have (81) 



If x = a. + if$ ; y = a ift then this equation reduces to 



v frV _ 



but as these are each positive terms their sum cannot be zero. This is a 

 particular case of 86. (See Appendix, note 2.) 



235. The limits of the roots. 



We can now show the limits between which the five roots, just proved to 

 be real, must actually lie in the equation 



