190 THE THEORY OF SCREWS. [190- 



shall acquire a given kinetic energy E, in consequence of the impulsive 

 wrench. 



We have from 91 the equation 



1 7/&quot; 2 



We can assign a geometrical interpretation to this equation, which will 

 lead to some interesting results. 



Through the centre of the pitch quadric the plane A reciprocal to 77 

 is to be drawn. A sphere (181) is to be described touching the plane A 

 at the origin 0, the diameter of the sphere being so chosen that the intercept 

 OP made by the sphere on a radius vector parallel to any screw 6 is equal 

 to ty^g (181). The quantity u e is inversely proportional to the radius vector 

 OQ of the ellipsoid of inertia, which is parallel to 9 ( 186). Hence for all 

 the screws of the screw system which acquire a given kinetic energy in 

 consequence of a given impulse, we must have the product OP . OQ constant. 



From a well-known property of the sphere, it follows that all the points 

 Q must lie upon a plane A , parallel to A. This plane cuts the ellipsoid of 

 inertia in an ellipse, and all the screws required must be parallel to the 

 generators of the cone of the second degree, formed by joining the points 

 of this ellipse to the origin, 0. 



Since we have already shown how, when the direction of a screw belonging 

 to a screw system of the third order is given, the actual situation of that 

 screw is determined ( 180), we are now enabled to ascertain all the screws 

 6 on which the body acted upon by a given impulse would acquire a given 

 kinetic energy. 



The distance between the planes A and A is proportional to OP . OQ, 

 and therefore to the square root of E. Hence, when the impulse is given, 

 the kinetic energy acquired on a screw determined by this construction is 

 greatest when A and A are as remote as possible. For this to happen, it 

 is obvious that A will just touch the ellipsoid of inertia. The group of 

 screws will, therefore, degenerate to the single screw parallel to the diameter 

 of the ellipsoid of inertia conjugate to A, But we have seen ( 130) that 

 the screw so determined is the screw which the body will naturally select 

 if permitted to make a choice from all the screws of the system of the 

 third order. We thus see again what Euler s theorem ( 94) would have 

 also told us, viz., that when a quiescent rigid body which has freedom of the 

 third order is set in motion by the action of a given impulsive wrench, the 

 kinetic energy which the body acquires is greater than it would have been 

 had the body been restricted to any other screw of the system than that 

 one which it naturally chooses. 



