232 THE THEORY OF SCREWS. [221- 



screw corresponding to those of a given impulsive screw can be deduced 

 from Euler s theorem ( 94). If a body receive an impulsive wrench on a 

 screw 7] while the body is constrained to twist about a screw 6, then we have 

 seen in 91 that the kinetic energy acquired is proportional to 



If #j, 0. 2 , 3 , #4 be the co-ordinates of referred to the four principal 

 screws of inertia belonging to the screw system of the fourth order, then 



(195,97) 



uf = uf + w;- &amp;lt;, 2 + u./* + ufOf. 



Hence we have to determine the four independent variables 1} 6.,, 3) # 4 , so 

 that 



shall be stationary. This is easily seen to be the case when lt 0.,, 3 , 4 are 

 respectively proportional to 



Jlfci *, |9t, ~^. 



U/l U 2 ^3 Ul 



These are accordingly, as we already know ( 97), the co-ordinates of the 

 screw about which the body will commence to twist after it has received 

 an impulsive wrench on ?/. 



This method might of course be applied to any order of freedom. 



222. General Remarks. 



It has been shown in 80 how the co-ordinates of the instantaneous 

 screw corresponding to a given impulsive screw can be determined when the 

 rigid body is perfectly free. It will be observed that the connexion between 

 the two screws depends only upon the three principal axes through the 

 centre of inertia, and the radii of gyration about these axes. We may 

 express this result more compactly by the well-known conception of the 

 momental ellipsoid. The centre of the momental ellipsoid is at the centre 

 of inertia of the rigid body, the directions of the principal axes of the 

 ellipsoid are the same as the principal axes of inertia, and the lengths of 

 the axes of the ellipsoid are inversely proportional to the corresponding 

 radii of gyration. When, therefore, the impulsive screw is given, the 

 momental ellipsoid alone must be capable of determining the corresponding 

 instantaneous screw. 



A family of rigid bodies may be conceived which have a common 



