82, 84, 85] EULER'S EQUATIONS. 231 



and then since H x = Ap, H y = Bq, H 2 = Cr, the equations become 

 simply 



30) 



These are Euler's dynamical equations for the rotation of a rigid body. 

 In case the moments of the applied forces about the origin 

 vanish, they become 



31) 



and we see that the quantities on the right, being the vector product 

 of the angular velocity by the angular momentum, represent the 

 centrifugal couple, which alone acts to produce the angular accel- 

 eration, whose components appear on the left. We thus obtain the 

 result obtained geometrically by Poinsot, the quantities on the left 

 denoting the velocity of the end of H in the ~body. 



The equations 29) may be simplified in another manner, if the 

 ellipsoid of inertia is of revolution. If for one of the moving axes 

 we take the axis of revolution, and for the others, any axes perpen- 

 dicular to it, whether fixed in the body or not, the axes will be 

 principal axes, and the moments of inertia constant, since the moment 

 of inertia about all axes perpendicular to the axis of rotation in the 

 same. Examples of this will be given in 96, 106. 



85. Poinsot's Discussion of the Motion. We may now 



integrate the equations 31) by making use of the fact that the 

 centrifugal couple is perpendicular to the angular velocity and the 

 angular momentum. Multiplying equations 31) respectively by p, q, r 

 and adding 



wHich is at once integrated as 



33) 4 Ap* + 4 Bf + } Cr> = const. 



This is the equation of energy. 



