50 THE PRINCIPAL SCREWS OF INERTIA. 



which expresses the freedom of the rigid body, n "screws 

 can be selected so that every pair of them are conjugate 

 screws of inertia (54). Let B ly &c. 7? 6 _ n be (6 - n) 

 screws defining the reciprocal screw-complex. Let A i be 

 any screw belonging to P. Then in the choice of A A l we 

 have n - i arbitrary quantities. Let /i be any impulsive 

 screw corresponding to A v as an instantaneous screw. 

 Choose A z reciprocal to /i, B ly &c. 7? 6 _ n , then AI and A 2 

 are conjugate screws, and in the choice of the latter we 

 have n- 2 arbitrary quantities. Let 7 2 be any impulsive 

 screw corresponding to A 2 as an instantaneous screw. 

 Choose A 3 reciprocal to I l9 7 2 , B l9 &c. 7? 6 _ M , and proceed 

 thus until A n has been attained, then each pair of the 

 group AU &c. A n are conjugate screws of inertia. The 

 number of quantities which remain arbitrary in the choice 

 of such a group amount to 



n (n - i) 



n 1+72 2 + &C. + I = ', 



2 



or exactly half the total number of arbitrary constants in 

 the selection of any n screws from a complex of the n th 

 order. 



57. Principal Screws of Inertia. It is the object of 

 this section to show that it is always possible to select 

 from the screw-complex of the n th order expressing the 

 freedom of a rigid body, one group of n screws, of which 

 every pair are both conjugate and reciprocal, and that 

 these constitute the principal screws of inertia ( 51). 



To prove this, it is sufficient to show that when the 

 remaining half of the arbitrary constants ( 56) have 

 been suitably disposed, then the group of n screws be- 

 sides being conjugate will be co-reciprocal. Choose A\ 

 reciprocal to B\ y &c. 7? 6 _ n , with n - i arbitrary quantities ; 

 A 2 reciprocal to Ai 9 B ly &c. B n -\, with n - 2 arbitrary 

 quantities, and so on, then the total number of arbitrary 



