DYNAMICS OF A RIGID BODY. 157 



#i0i + #202 + &C. + a n O n = O. 



All the screws whose co-ordinates satisfy this equa- 

 tion must be reciprocal to the screw belonging to A, of 

 which the co-ordinates are proportional to 



#! a n ^ 



/l ' pn* 



hence all the screws whose co-ordinates satisfy the 

 linear equation must be reciprocal to 7 - n independent 

 screws, viz., % and 6 - n screws from the screw complex 

 reciprocal to A . Hence we have the following theorem 

 ( 46). 



If from a screw complex (A) of the n th order and first 

 degree, we select all the screws whose n co-ordinates 

 (when referred to n screws of reference belonging to A) 

 satisfy one linear equation, then the group of screws so 

 selected constitute a screw complex of the (n - i} th order 

 and first degree. 



We shall now define a screw complex of the (n- i) th 

 order and second degeee. 



If from a screw complex A of the n th order and 

 first degree, we select all the screws whose n co-ordi- 

 nates (when referred to n screws of reference belonging 

 to A) satisfy one homogeneous equation of the second 

 degree, then the group of screws so selected constitute a 

 screw complex of the (n - i) th order and second degree. 



147. Polar Screws. Let U = o denote a screw com- 

 plex of the (n - i) th order and second degree, embraced 

 within the screw complex of the n th order and first 

 degree, which is denoted by A, then we define the polar 

 of the screw with respect to U Q = o to be the screw 

 belonging to A, of which the n co-ordinates are propor- 

 tional to 



