Let us suppose a system of particles moving under the influence of their mutual actions. Let O be any point, and draw any plane L through O. Then the moment of momentum of the system around the point O and projected into the plane L is constant. Let us call it S. If another plane, L´, had been drawn through O, the similar moment with regard to L´ is S´. Thus for each plane through O there will be a corresponding value of S. We have now to show that one plane can be drawn through O, such that the value of S is greater than it is for any other plane. This is the principal plane of the system. If _v_ be the velocity of a particle, then in a small time _t_ it moves over the distance _v t_. If _p_ be the perpendicular from O on the tangent to the motion, then the area of the triangle swept round O in the time _t_ is ½ _p v t_, and we see that the momentum is proportional to the mass of the particle multiplied into the area swept over in the time _t_. The quantity S will, therefore, be proportional to the sum of the projections of the areas in L, swept over in the time _t_, each increased in the proportion of the mass of the particle. It is easily seen that the projection of an area in one plane on another is obtained by multiplying the original area by the cosine of the angle between the two planes. For if the area be divided into thin strips by lines parallel to the line of intersection of the planes, then in the projection of these strips the lengths are unchanged, while the breadths are altered by being multiplied by the cosine of the angle between the two planes. If, therefore, we mark off on the normal to a plane L a length _h_ proportional to any area in that plane, then the projection of this area on any other plane L´ may be measured by the projection of _h_ on the normal to L´. [Illustration: Fig. 63.—MOMENT OF MOMENTUM UNALTERED BY COLLISION.] To determine the moment of momentum resolved in any plane we therefore proceed as follows: Draw a plane through O, and the tangent to the path of one of the particles, and mark off on the normal drawn through O to this plane a length _l_ proportional to the moment of momentum. Repeat the same process for each of the other particles with lengths _l´_, _l″_, etc., on their several normals. Suppose that _l_, _l´_, _l″_ represent forces acting at O, and determine their resultant R. Then R, resolved along any other direction, will give the component of moment of momentum in the plane to which that direction is normal. In any plane which passes through R the component of moment of momentum is zero. The plane perpendicular to R contains the maximum projection of moment of momentum. This is the principal plane of the system which we have seen to be of such importance in connection with the nebular theory. § 15. COLLISIONS.