225.] GENERAL PRINCIPLES. I2 3 



of the particle m about the axis of x. We may now agree to 

 call the quantity ^m(yz-zy) the ang^ilar momentum of the 

 body about the axis of x> just as ^mx is the linear momentum 

 of the body along this axis ; and similarly for the other axes. 

 The meaning of the equations (7) can then be stated as follows : 

 The rate at which the angular momentum of a rigid body about 

 any axis changes with the time is equal to the sum of the 

 moments of all the external forces about this line. 



The equations (4) and (6), or (5) and (7), are the six equa- 

 tions of motion of the rigid body. The three equations (4) or 

 (5) may be called the equations of linear momentum, while (6) 

 or (7) are the equations of angular momentum. 



225. The equations (4) and (6) can also be derived from the equa- 

 tion (3), which expresses d'Alembert's principle, by selecting for #, 

 By, Bz convenient displacements. 



Thus, the rigidity of the body will evidently not be disturbed if we 

 give to all its points equal and parallel infinitesimal displacements, since 

 this merely amounts to subjecting the whole body to an infinitesimal 

 translation. Equation (3) can in this case be written 



&#:(- mx + X) + 8y2( my+ Y) + Bz%( mz +Z) = o, 



and is therefore equivalent to the three equations (4), since 8.*, By, Bz 

 are independent and arbitrary. 



Again, let the body be subjected to an infinitesimal rotation of angle 

 BO about any line /. 



As shown in Art. 293 of Part I., the linear velocities of any point 

 (x, y, z) of a rigid body, due to a rotation of angular velocity o> = 80/8/ 

 about any line / are, if eo,., o> y , w z denote the components of o> : 



x = o)yZ <a z y, y = (DgX <D X Z, z = <D x y oyr. 



Hence, putting eo x 8/=80 x , o> y S/=80 y , <u,8/=80,, we have for the 

 displacements of the point (x, y, z) , due to a rotation of angle 80, 



Bx = zBO y -yBO z , By = xBe z -zB9 x , Bz = y S0 Z - x W r 



If these values be introduced into d'Alembert's equation (3) and the 

 terms in S0 Z , B0 y , W z be collected, it assumes the form 



S0 X 2[- m(yz - zy) +yZ-zY] + S0 y 2[- m(zx - xz) + zX- xZ~\ 

 + 80,5 [ m(xy yx) + xYyX']=o ; 



