This important theorem is deduced from the fact stated in the third law of motion, that action and reaction are equal and opposite. Let us take any two particles; then, the acceleration of the moment of momentum of one of them, A, by the action of the other, B, will be the moment of the force between them. The acceleration of the moment of momentum of B by the action of A will be the same moment, but with an opposite sign. Hence the total acceleration of the moment of momentum of the system by the mutual action of A and B is zero. In like manner we dispose of every other pair of actions, and thus, as the total acceleration of the moment of momentum is zero, it follows that the moment of momentum of the system itself must be constant. This fundamental principle is also known as the doctrine of the conservation of areas. It may be stated in the following manner:— _If a system of particles are moving in a plane under the influence of their mutual actions only, the algebraic sum of the areas swept out around a point, each multiplied by the mass of the particle, is directly proportional to the time._ § 13. IF A PARTICLE OF MASS _m_, IS MOVING IN SPACE UNDER THE ACTION OF ANY FORCE F, THEN THE PROJECTION OF THAT PARTICLE ON ANY FIXED PLANE WILL MOVE AS IF IT WERE A PARTICLE OF MASS _m_ ACTED UPON BY THAT COMPONENT OF F WHICH IS PARALLEL TO THE PLANE. This is evident from the consideration that the acceleration of the particle parallel to the plane must be proportional to this component of F. Let us now suppose a system of particles moving in space under their mutual actions. The projections of these particles on a plane will move as if they were the particles themselves subjected to the action of forces which are the projections of the actual forces on the same plane, and as the reactions between any two particles are equal and opposite, the projections of those reactions on the plane are equal and opposite. Hence the proof already given of the constancy of the moments of momentum of a plane system, will apply equally to prove the constancy of the moments of momentum of the projections of the particles on the plane. Hence we have the following important theorem:— _Let a system of particles be moving in space under the action of forces internal to the system only. Let any plane be taken, and any point in that plane, and let the momentum of each particle be projected into the plane, then the algebraic sum of the moments of these projections around the point is constant._ § 14. ON THE PRINCIPAL PLANE OF A SYSTEM.