Article 59. Definition of a mass-vector
We have seen that a vector represents the operation of carrying a tracing point from a given origin to a given point.
Let us define a mass-vector as the operation of carrying a given mass from the origin to the given point. The direction of the mass-vector is the same as that of the vector of the mass, but its magnitude is the product of the mass into the vector of the mass.
Thus if OA is the vector of the mass A, the mass-vector is OA • A.

Article 60. Centre of mass of two particles
If A and B are two masses, and if a point C be taken in the straight line AB, so that BC is to CA as A to B, then the mass-vector of a mass A + B placed at C is equal to the sum of the mass-vectors of A and B.

Figure 7

For OA • A + OB • B
= (OC + CA)A + (OC + CB)B.
= OC(A + B) + CA • A + CB • B.
Now the mass-vectors CA A and CB B are equal and opposite, and so destroy each other, so that OA • A + OB • B = OC(A + B) or, C is a point such that if the masses of A and B were concentrated at C, their mass-vector from any origin O would be the same as when A and B are in their actual positions. The point C is called the centre of mass of A and B.

Article 61. Centre of mass of a system
If the system consists of any number of particles, we may begin by finding the centre of mass of any two particles, and substituting for the two particles a particle equal to their sum placed at their centre of mass. We may then find the centre of mass of this particle, together with the third particle of the system, and place the sum of the three particles at this point, and so on till we have found the centre of mass of the whole system.
The mass-vector drawn from any origin to a mass equal to that of the whole system placed at the centre of mass of the system is equal to the sum of the mass-vectors drawn from the same origin to all the particles of the system.
It follows, from the proof in Article 60, that the point found by the construction here given satisfies this condition. It is plain from the condition itself that only one point can satisfy it. Hence the construction must lead to the same result, as to the position of the centre of mass, in whatever order we take the particles of the system.
The centre of mass is therefore a definite point in the diagram of the configuration of the system. By assigning to the different points in the diagrams of displacement, velocity, total acceleration, and rate of acceleration, the masses of the bodies to which they correspond, we may find in each of these diagrams a point which corresponds to the centre of mass and indicates the displacement, velocity, total acceleration, or rate of acceleration of the centre of mass.

Article 62. Momentum represented as the rate of change of a mass-vector
In the diagram of velocities, if the points o, a, b, c correspond to the velocities of the origin O and the bodies A, B, C, and if p be the centre of mass of A and B placed at a and b respectively, and if q is the centre of mass of A + B, placed at p and C at c, then q will be the centre of mass of the system of bodies A, B, C at a, b, c, respectively.

Figure 8

The velocity of A with respect to O is indicated by the vector oa, and that of B and C by ob and oc.op is the velocity of the centre of mass of A and B, and oq that of the centre of mass of A, B, and C, with respect to O.
The momentum of A with respect to O is the product of the velocity into the mass, or oa A, or what we have already called the mass-vector, drawn from o to the mass A at a. Similarly the momentum of any other body is the mass-vector drawn from o to the point on the diagram of velocities corresponding to that body, and the momentum of the mass of the system concentrated at the centre of mass is the mass-vector drawn from o to the whole mass at q.
Since, therefore, a mass-vector in the diagram of velocities is what we have already defined as a momentum, we may state the property proved in Article 61, in terms of momenta, thus: The momentum of a mass equal to that of the whole system, moving with the velocity of the centre of mass of the system, is equal in magnitude and parallel in direction to the sum of the momenta of all the particles of the system.

Article 63. Effect of external forces on the motion of the centre of mass

Figure 9

In the same way in the diagram of total acceleration the vectors wa, wb, drawn from the origin, represent the change of velocity of the bodies A, B, etc., during a certain interval of time. The corresponding mass-vectors, wa A, wb B, etc., represent the corresponding changes of momentum, or, by the second law of motion, the impulses of the forces acting on these bodies during that interval of time. If k is the centre of mass of the system, wk is the change of velocity during the interval, and wk(A + B + C) is the momentum generated in the mass concentrated at the centre of gravity. Hence, by Article 61, the change of momentum of the imaginary mass equal to that of the whole system concentrated at the centre of mass is equal to the sum of the changes of momentum of all the different bodies of the system.
In virtue of the second law of motion we may put this result in the following form:
The effect of the forces acting on the different bodies of the system in altering the motion of the centre of mass of the system is the same as if all these forces had been applied to a mass equal to the whole mass of system, and coinciding with its centre of mass.

Article 64. The motion of the centre of mass of a system is not affected by the mutual action of the parts of the system.
For if there is an action between two parts of the system, say A and B, the action of A on B is always, by the third law of motion, equal and opposite to the reaction of B on A. The momentum generated in B by the action of A during any interval is therefore equal and opposite to that generated in A by the reaction of B during the same interval, and the motion of the centre of mass of A and B is therefore not affected by their mutual action.
We may apply the result of the last article to this case and say that, since the forces on A and on B arising from their mutual action are equal and opposite, and since the effect of these forces on the motion of the centre of mass of the system is the same as if they had been applied to a particle whose mass is equal to the whole mass of the system, and since the effect of two forces equal and opposite to each other is zero, the motion of the centre of mass will not be affected.

Article 65. First and second laws of motion
This is a very important result. It enables us to render more precise the enunciation of the first and second laws of motion, by defining that by the velocity of a body is meant the velocity of its centre of mass. The body may be rotating, or it may consist of parts, and be capable of changes of configuration, so that the motions of different parts may be different, but we can still assert the laws of motion in the following form:
Law I. The centre of mass of the system perseveres in its state of rest, or of uniform motion in a straight line, except insofar as it is made to change that state by forces acting on the system from without.
Law II. The change of momentum of the system during any interval of time is measured by the sum of the impulses of the external forces during that interval.

Article 66. Method of treating systems of molecules
When the system is made up of parts which are so small that we cannot observe them, and whose motions are so rapid and so variable that even if we could observe them we could not describe them, we are still able to deal with the motion of the centre of mass of the system, because the internal forces which cause the variation of the motion of the parts do not affect the motion of the centre of mass.

Article 67. By the introduction of the idea of mass we pass from point-vectors, point displacements, velocities, total accelerations, and rates of acceleration, to mass-vectors, mass displacements, momenta, impulses, and moving forces.
In the diagram of rates of acceleration (fig. 9, Art. 63) the vectors wa, wb, etc., drawn from the origin, represent the rates of acceleration of the bodies A, B, etc., at a given instant, with respect to that of the origin O.
The corresponding mass-vectors, wa A, wbB, etc., represent the forces acting on the bodies A, B, etc.
We sometimes speak of several forces acting on a body, when the force acting on the body arises from several different causes, so that we naturally consider the parts of the force arising from these different causes separately.
But when we consider force, not with respect to its causes, but with respect to its effect—that of altering the motion of a body—we speak not of the forces but of the force acting on the body, and this force is measured by the rate of change of the momentum of the body and is indicated by the mass-vector in the diagram of rates of acceleration.
We have thus a series of different kinds of mass-vectors corresponding to the series of vectors which we have already discussed.
We have, in the first place, a system of mass-vectors with a common origin, which we may regard as a method of indicating the distribution of mass in a material system, just as the corresponding system of vectors indicates the geometrical configuration of the system.
In the next place, by comparing the distribution of mass at two different epochs, we obtain a system of mass-vectors of displacement.
The rate of mass displacement is momentum, just as the rate of displacement is velocity.
The change of momentum is impulse, as the change of velocity is total acceleration.
The rate of change of momentum is moving force, as the rate of change of velocity is rate of acceleration.

Article 68. Definition of a mass-area
When a material particle moves from one point to another, twice the area swept out by the vector of the particle multiplied by the mass of the particle is called the mass-area of the displacement of the particle with respect to the origin from which the vector is drawn.
If the area is in one plane, the direction of the mass-area is normal to the plane, drawn so that, looking in the positive direction along the normal, the motion of the particle round its area appears to be the direction of the motion of the hands of a watch.
If the area is not in one plane, the path of the particle must be divided into portions so small that each coincides sensibly with a straight line, and the mass-areas corresponding to these portions must be added together by the rule for the addition of vectors.

Article 69. Angular momentum
The rate of change of a mass-area is twice the mass of the particle into the triangle, whose vertex is the origin and whose base is the velocity of the particle measured along the line through the particle in the direction of its motion. The direction of this mass-area is indicated by the normal drawn according to the rule given above.
The rate of change of the mass-area of a particle is called the angular momentum of the particle about the origin, and the sum of the angular momenta of all the particles is called the angular momentum of the system about the origin.
The angular momentum of a material system with respect to a point is, therefore, a quantity having a definite direction as well as a definite magnitude.
The definition of the angular momentum of a particle about a point may be expressed somewhat differently as the product of the momentum of the particle with respect to that point into the perpendicular from that point on the line of motion of the particle at that instant.

Article 70. Moment of a force about a point
The rate of increase of the angular momentum of a particle is the continued product of the rate of acceleration of the velocity of the particle into the mass of the particle into the perpendicular from the origin on the line through the particle along which the acceleration takes place. In other words, it is the product of the moving force acting on the particle into the perpendicular from the origin on the line of action of this force.
Now the product of a force into the perpendicular from the origin on its line of action is called the moment of the force about the origin. The axis of the moment, which indicates its direction, is a vector drawn perpendicular to the plane passing through the force and the origin, and in such a direction that, looking along this line in the direction in which it is drawn, the force tends to move the particle round the origin in the direction of the hands of a watch.
Hence the rate of change of the angular momentum of a particle about the origin is measured by the moment of the force which acts on the particle about that point.
The rate of change of the angular momentum of a material system about the origin is in like manner measured by the geometric sum of the moments of the forces which act on the particles of the system.

Article 71. Conservation of angular momentum
Now consider any two particles of the system. The forces acting on these two particles, arising from their mutual action, are equal, opposite, and in the same straight line. Hence the moments of these forces about any point as origin are equal, opposite, and about the same axis. The sum of these moments is therefore zero. In like manner the mutual action between every other pair of particles in the system consists of two forces, the sum of whose moments is zero.
Hence the mutual action between the bodies of a material system does not affect the geometric sum of the moments of the forces. The only forces, therefore, which need be considered in finding the geometric sum of the moments are those which are external to the system—that is to say, between the whole or any part of the system and bodies not included in the system.
The rate of change of the angular momentum of the system is therefore measured by the geometric sum of the moments of the external forces acting on the system.
If the directions of all the external forces pass through the origin, their moments are zero, and the angular momentum of the system will remain constant.
When a planet describes an orbit about the sun, the direction of the mutual action between the two bodies always passes through their common centre of mass. Hence the angular momentum of either body about their common centre of mass remains constant, so far as these two bodies only are concerned, though it may be affected by the action of other planets. If, however, we include all the planets in the system, the geometric sum of their angular momenta about their common centre of mass will remain absolutely constant, whatever may be their mutual actions, provided no force arising from bodies external to the whole solar system acts in an unequal manner upon the different members of the system.