303.] THE RIGID BODY. j^j 



of changing the magnitude of o> by the amount da, without 

 affecting the direction of the axis, while the effect of the com- 

 ponent a)U is to incline the axis / by the angle dcr. 



303. To obtain analytical expressions for the components of 

 the acceleration of any point P of a rigid body in spherical 

 motion, let us take the centre of rotation O as origin of a 

 system of fixed rectangular axes. Let x, y, z be the co-ordinates 

 of P \ a, @, 7 the direction cosines of the instantaneous axis /; 

 and X, fji, v those of the perpendicular PQ = r let fall from P on 

 this axis /. 



The total acceleration of P is composed of the centripetal 

 acceleration o>V, which is directed along PQ, and the component 

 arising from the angular acceleration a (Art. 301). 



The components of o>V along the axes of x, y, z are \&>V, 

 /xcoV, vco 2 r. Projecting the closed polygon OQPO on each of the 

 axes, we find 



or, since OQ is the projection of OP on /, i.e. OQ = 

 \r= a (ax 



vr=y (ax + fty + yz) z. 



Multiplying these equations by o> 2 and putting aco = co x , /3o) = co yt 

 jo) = a) z , we find for the components of the centripetal accelera- 

 tion of the point (x, y, z) : 



= co x a) x 

 G) 2/ (w x ^H-ft>yj+G>^') oPy* (8) 



= co z (w x x -f- co y y + &>^) aPz. 



The angular acceleration a = d$/dt (Art. 301) has for its 

 components along the axes of x, y, z 



d> x _ d(0 y _ dto, 



CC = - -f ** - *> ^z - 



* dt dt dt 



