236 MOTION OF A RIGID BODY IN TWO DIMENSIONS. [CHAP. XI. 



Let the forces acting on the body be reduced to a resultant 

 force at its centre of inertia and a couple. Let P, Q be the 

 resolved parts of the force in the directions in which the accelera- 

 tion of the centroid was resolved, and let N be the couple. 



Then the system of vectors expressed by Mf l} Mf 2) Mk z & has 

 the same resolved part in any direction, and the same moment 

 about any axis, as the system P, Q, N. 



In particular we have 



and the equations of motion of the body can always be written in 

 this form. 



In the formation of equations of motion diversity can arise 

 from the choice of directions in which to resolve, and of axes 

 about which to take moments. As in the case of Dynamics of a 

 Particle, the equations arrived at are differential equations, and 

 no rules can be given for solving them in general. If however 

 the circumstances are such that there is an equation of energy, or 

 an equation of conservation of momentum, such equations are first 

 integrals of the equations of motion. 



222. Continuance of motion in two dimensions. The 



question arises whether a body, which at some instant is moving 

 in two dimensions parallel to a certain plane, continues to move 

 parallel to that plane or will presently be found to be moving in a 

 different manner. A general answer to this question cannot be 

 given here, but it is clear that there is a class of cases in which 

 the motion in two dimensions persists. Thus this class includes 

 all the cases in which the body is symmetrical with respect to a 

 plane and the forces applied to it are directed along lines lying in 

 that plane, or, more generally, when the forces can be reduced to 

 a single resultant in the plane of symmetry and a couple about an 

 axis perpendicular to that plane. 



223. Rigid Pendulum*. A heavy body free to rotate 

 about a fixed horizontal axis is known as a " compound pendulum" 



* Huygens was the first to solve the problem of the motion of the pendulum, 

 and the principles which he invoked were among the considerations which ulti- 

 mately led to the establishment of the Theory of Energy. 



