46 THE PRINCIPAL SCREWS OF INERTIA. 



while in the chapters on the special orders of freedom we 

 shall show how the principal screws of inertia are to be 

 determined for each case. 



52. Screws of Reference. We have now to define the 

 group of six co-reciprocal screws ( 2 8) which are pecu- 

 liarly adapted to serve as the screws of reference in 

 Kinetic investigations. Let O be the centre of inertia of 

 the rigid body, and let OA, OB, OC be the three prin- 

 cipal axes through O, while a, b, c are the corresponding 

 radii of gyration. Then two screws along OA, viz. : w : , 

 W2, with pitches + a, - a ; two screws along OB, viz. : o> 3 , 

 <>4, with pitches + b, - b, and two along OC, viz. : o> 5 , W G , 

 with pitches + c, - c, are the co-reciprocal group which we 

 shall employ. For convenience in writing the formulae, 

 we shall often use / &c. p Q , to denote the pitches as 

 before. 



We shall now prove that the six screws thus defined 

 are the principal screws of inertia of a free rigid body. 

 Let the mass of the body be M, and let an impulsive 

 wrench on wi act for a short time /. The intensity of 

 this wrench is tui", and the moment of the couple is a^ 1 . 

 We now consider the effect of the two portions of the 

 wrench separately. The effect of the force wi" is to give 

 the body a velocity of translation parallel to OA and 



equal to -^. wi ". By the property of the principal axes 



the effect of the couple will be to impart an angular ve- 

 locity w/ about the axis OA. This angular velocity 

 is easily determined. The effective force which must 

 have acted upon a particle dm at a perpendicular dis- 



tance r from OA is -~ dm. The sum of the moments 





of all these forces is Ma? ^ . This quantity must equal 



