CHAPTER XXIII. 



VARIOUS EXERCISES. 



321 The Co-ordinates of a Rigid Body. 



We have already explained ( 302) how nine co-ordinates define a rigid 

 body sufficiently for the present theory. One set of such co-ordinates with 

 respect to any three rectangular axes may be obtained as follows. 



Let the element dm have the co-ordinates x, y, z, then causing the 

 integrals to extend over the whole mass, we compute the nine quantities 



fxdm = Mx ; fydm = My ; fzdm = Mz ; 



jyzdm = Ml? ; fxzdm = M1 2 2 ; fay dm = M1 3 2 ; 



j(y* + z 2 ) dm = Mp, 2 ; JO 2 + z 2 ) dm = Mp 2 2 ; f(x 2 + f) dm = Mp 3 2 . 



The nine quantities #, y , z , I?, 1 2 2 , 1 3 2 , pf, p.?, p/ constitute an adequate 

 system of co-ordinates of the rigid body. 



If 0J, 2 , ... 6 S be the canonical co-ordinates of a screw about which twists 

 a rigid body whose co-ordinates are x , y , z , If, If, 1 3 2 , pf, pf, p 3 * with respect 

 to the associated Cartesian axes, then the kinetic energy is Mu e 2 d 2 , where M 

 is the mass, $ the twist velocity, and where 



V = a?e? + a?G? + b 2 3 2 + W + c 2 6&amp;gt; 5 2 + c 2 6 2 



+ 6^0 (^3 - #4) (0* + 0*) ~ ex. (0 5 - 6 ) (0 3 + t ) 

 + cy (6, - 6 ) (0, + 6,} - ay, (0, - 0,) (0, + 0.) 

 + az (0, - 2 ) (0 3 + 0.) - bz (0 3 - 4 ) (0, + 6&amp;gt; 2 ) 

 + i (/&amp;gt;! 2 - 2 ) (0 l + &amp;lt;9 2 ) 2 + i (pj - &) (0 t + 6tf + \ (p 3 2 - c 2 ) (0. + 6 ) 2 



- V (0* + 4 ) (O, + 0.) ~ ^ (0* + 0.) (0i + 0) - tt (0i + 0*) (0* + 04)- 



232 



