260 THE THEORY OF SCREWS. [239- 



If a, b, c be the radii of gyration, then the instantaneous screw cor 

 responding to i] has for co-ordinates 



, % _fh ,Va _ r h ,^5 _^s 



I y y I T &amp;gt; 71 I i 



a a o o c c 



The condition that 77 and its instantaneous screw shall be parallel to a 

 pair of conjugate diameters of the momental ellipsoid is 



or 



But if the impulsive wrench on 77 be a force, then the pitch of 77 is zero, 

 whence the theorem is proved. 



240. Theorem. 



When an impulsive wrench acting on a free rigid body produces an 

 instantaneous rotation, the axis of the rotation must be perpendicular to 

 the impulsive screw. 



Let i)!, ... i) 6 be the axis of the rotation, then 



2^ V = 0, 

 or 



a (77! - 7; 2 ) (77! + O + b (773 - 774) (773 + 774) + c (775 - 77 6 ) (775 + 77 6 ) = 0, 



whence the screw of which the co-ordinates are + ar) l , arj. 2 , + brj 3 , ... is 

 perpendicular to 77, and the theorem is proved. 



From this theorem, and the last, we infer that, when an impulsive force 

 acting on a rigid body produces an instantaneous rotation, the direction of 

 the force, and the axis of the rotation, are parallel to the principal axes of 

 a section of the momental ellipsoid. 



241. Principal Axis. 



If 77 be a principal axis of a rigid body, it is required to prove that 



SjH*y0, 

 reference being made to the absolute principal screws of inertia. 



For in this case a force along a line Q intersecting 77, compounded with 

 a couple in a plane perpendicular to 77, must constitute an impulsive wrench 

 to which 77 corresponds as an instantaneous screw, whence we deduce ( 120), 

 h and k being the same for each coordinate, 



., h dR 



h dR 



