84] THE PRINCIPAL SCREWS OF INERTIA. 73 



so that every pair of them are conjugate screws of inertia (81). Let B l , &c. 

 B 6 _ n be (6 n) screws defining the reciprocal screw system. Let A l be any 

 screw belonging to P. Then in the choice of A l we have n 1 arbitrary 

 quantities. Let 7 X be any impulsive screw corresponding to A l as an instan 

 taneous screw. Choose A 2 reciprocal to 7 1( B 1} ... B s ^ n , then A l and A 2 are 

 conjugate screws, and in the choice of the latter we have n 2 arbitrary 

 quantities. Let /., be any impulsive screw corresponding to A 2 as an instan 

 taneous screw. Choose A 3 reciprocal to I I , I 2 , B l} ... B 6 _ n , and proceed thus 

 until A n has been attained, then each pair of the group A l} &c. A n are 

 conjugate screws of inertia. The number of quantities which remain 

 arbitrary in the choice of such a group amount to 



or exactly half the total number of arbitrary constants disposable in the 

 selection of any n screws from a system of the nth order. 



84. Principal Screws of Inertia. 



We have now to prove the important theorem in Dynamics which affirms 

 the existence of n principal Screws of Inertia in a rigid body with n degrees 

 of freedom. 



The proof that we shall give is, for the sake of convenience, enunciated 

 with respect to the freedom of the third order, but the same method applies 

 to each of the other degrees of freedom. 



Let 6 be one of the principal screws of inertia, then an impulsive wrench 

 on must make the body commence to twist about 0. In the most general 

 case when the body is submitted to constraint, the impulsive wrench on will 

 of course be compounded with the reaction on some screw A, of the reciprocal 

 system. The result will be to produce the impulsive wrench which would, 

 if the body had been free, have generated an instantaneous twist velocity 

 about 6. 



We thus have the following equations ( 80) where x and y are unknown : 



pA = uQi + yXi, 



Let a be one of the screws of the three-system in question. Then since X 

 must be reciprocal to a we have by multiplying these equations respectively 

 by pii, . .. p s ct 6 and adding, 



K = xp^e, + xp.a.,6, + . . . + xp ( &$. 



