70 THE THEORY OF SCREWS. [79- 



79. Screws of Reference. 



We have now to define the group of six co-reciprocal screws ( 31) which 

 are peculiarly adapted to serve as the screws of reference in Kinetic investi 

 gations. Let be the centre of inertia of the rigid body, and let OA, OB, 

 OC be the three principal axes through 0, while a, b, c are the corresponding 

 radii of gyration. Then two screws along OA, viz. w l , &&amp;gt; 2 , with pitches + a, 

 a; two screws along OB, viz. o&amp;gt; 3 , &&amp;gt; 4 , with pitches + b, b, and two along 

 OC, viz. &amp;lt;y D , w 6) with pitches + c, c, are the co-reciprocal group which 

 we shall employ. The group thus indicated form of course a set of canonical 

 co-reciprocals ( 41). For convenience in writing the formulae, we shall 

 often use p ly ... p 6 , to denote the pitches as before. 



We shall first prove that the six screws thus defined are the principal screws 

 of inertia of the rigid body when perfectly free. Let the mass of the body be 

 M, and let a great constant wrench on w, act for a short time e. The intensity 

 of this wrench is &&amp;gt;/ , and the moment of the couple is aw&quot;. We now consider 

 the effect of the two portions of the wrench separately. The effect of the force 

 &&amp;gt;/ is to give the body a velocity of translation parallel to OA and equal to 



/? 



-rj / . The effect of the couple is to impart an angular velocity d^ about 



the axis OA. This angular velocity is easily determined. The effective 

 force which must have acted upon a particle dm at a perpendicular distance 



r from OA is - dm. The sum of the moments of all these forces is 

 e 



Mo? -~ . This quantity is equal to the moment of the given couple so that 



60 



Ma 2 - 1 = aw,&quot;, 



e 



&&amp;gt; x 



The total effect of the wrench on w l is, therefore, to give the body a 

 velocity of translation parallel to OA, and equal to T^WI&quot;, and also a velocity 



ff 



of rotation about OA equal to ^ &&amp;gt;/ . These movements unite to form a 



twisting motion about a screw on OA, of which the pitch, found by dividing 

 the velocity of translation by the velocity of rotation, is equal to a. This 

 same quantity is however the pitch of eoj, and thus it is proved that an 

 impulsive wrench on ^ will make the body commence to twist about w^. 

 We shall in future represent ew&quot; by the symbol &&amp;gt;/&quot;, which is accordingly to 

 express the intensity of the impulsive wrench. 



