128 KINETICS OF A RIGID BODY. [233. 



where the quantity in parenthesis is the cosine of the angle 

 between the directions (', /3', 7') and (a, (3, 7), and is therefore 

 greatest when these directions coincide. 



The plane (12) about whose normal the angular momentum 

 is greatest, and by projection on which the area 5 is made 

 greatest, is called Laplace's invariable plane. As its equation is 

 independent of /, it remains fixed. The normal of this plane is 

 sometimes called the invariable line or direction. 



233. Let us now return to the general case of the motion of 

 a rigid body acted upon by any forces whatever. 



The propositions of Arts. 226 and 229 together establish the 

 so-called principle of the independence of the motions of translation 

 and rotation. In studying the motion of a rigid body it is 

 possible, according to this principle, to consider separately the 

 motion of translation of the centroid, and the rotation of the 

 body about the centroid. 



By Art. 226, the motion of the centroid is the same as that 

 of a particle of mass M acted upon by all the external forces 

 transferred parallel to themselves to the centroid. As the 

 motion of a particle has been discussed in Chapter V., nothing 

 further need be said about this part of the problem. 



By Art. 229, the motion of the body about the centroid is the 

 same as if the centroid were fixed. The problem of the motion 

 of a rigid body with a fixed point is therefore of great impor- 

 tance ; it will be discussed in Section IV. The more simpl< 

 special case of a rigid body with a fixed axis is treated in Sec- 

 tion III. The solution of both these problems depends on the 

 equations (6) or (7). 



234. In d'Alembert's equation (3) it is of course allowable to 

 substitute for the virtual displacements %x, 8y, &z the actual dis- 

 placements dx, dy, dz of the particles in any motion of a fre< 

 rigid body, since these actual displacements are certainly com- 

 patible with the condition of rigidity. The equation can thei 

 be written 



(xdx +ydy + zdz) = 2 (Xdx +Ydy + Zde). (13) 



