1 14 DYNAMICS OF A RIGID BODY. 



the six absolute principal screws of inertia* as screws of 

 reference, we must have ( 66) 



dR 



' 



where 17 denotes the screw of zero pitch on the same 

 straight line. 



Expanding this equation, and reducing, we find 



This result can be much simplified. By comparing 

 | 37 and 52, it appears that 



R = (t|i + J? 2 ) 2 + (i?s + i?0* + (?5 + le) 2 - 



and therefore 



dR . 



S/ilJi -T = 22/irji 2 = 2/rj = O. 



Hence we can prove that the product of the pitches of two 

 conjugate screws of inertia is constant, and is equal to 

 minus the square of the radius of gyration about the 

 common axis of the screws. 



1 06. Reaction of Constraints. We shall now con- 

 sider the following problem : A body which is free to 

 twist about all the screws of a cylindroid C receives 

 an impulsive wrench on a certain screw tj. It is re- 

 quired to find the screw X, a wrench on which con- 

 stitutes the impulsive reaction of the constraints. Let 

 C represent the cylindroid which, if the body were per- 

 fectly free, would form the locus of those screws, impul- 



* We shall often find it convenient to designate the six principal screws of 

 inertia of a free rigid body ( 52) by the phrase dbolute principal screws of 

 inertia. 



