227] FREEDOM OF THE FOURTH ORDER. 241 



Hence we deduce that the screw with co-ordinates 



i, Oj) 6&amp;gt; 



and the screw with co-ordinates proportional to 



.1 d JL 1 ^ 1 ^ 



p l d^ p 2 da z p 6 dot G 

 in which U is the expression 



P^ + p 2 2 2 ... + p 6 a&amp;lt;? + X (i 2 + 02 2 ... + 2 ai a 2 cos (12) ...) 

 must be collinear, and this is true for all values of X. 



We hence see that the co-ordinates of a screw collinear with a must be 

 proportional to 



where 



E = a 1 2 + a 2 2 +... + 2a 1 a 2 cos (12) + ... 



Thus we obtain the results of 47 in a different manner. 



227. Dynamical application of Polar Screws. 



We have seen ( 97) that the kinetic energy of a body twisting about a 



7/n/ 



screw 6 with a twist velocity -^- and belonging to a w-system is 



w 



the screws of reference being the principal screws of inertia. 



If we make i&amp;lt; 1 2 1 2 + ... + u n 2 n 2 = 0, then 6 must belong to a quadratic 

 -system. This system is, of course, imaginary, for the kinetic energy of 

 the body when twisting about any screw which belongs to it is zero*. 



The polar 77 of the screw 0, with respect to this quadratic w-system, has 

 co-ordinates proportional to 



u i n u n / 



U 1} ... p. 



Pi Pn 



Comparing this with 97, we deduce the following important theorem : 



A quiescent rigid body is free to twist about all the screws of an enclosing 

 (n + \}-system A. If the body receive an impulsive wrench on a screw 17 



* In a letter to the writer, Professor Klein pointed out many years ago the importance of the 

 above screw system. He was led to it by expressing the condition that the impulsive screw 

 should be reciprocal to the corresponding instantaneous screw. 



B. 16 



