AND DETERMINANTS OF LINES. 
495 
Now, by a well-known proposition, it is evident that S(w 2 R)=:MR, where M is the mass 
of the body and II is the radius of the body’s centre of gravity. Hence 
U=MD,(R); 
and therefore, if V denote the velocity of the centre of gravity in magnitude and direction, 
U=MV, or, in other words, U, the body's momentum, is the momentum of the body’s mass 
collected at the body's centre of gravity. 
54. In the next place we have to find H. The investigation will be very much 
facilitated by the following consideration. A body’s motion is said to be compounded 
of motions a, (3, y, if the velocity of each of the body’s particles may be considered as 
the resultant of the respective velocities due to the motions of a, j3, y separately. In 
such case the momentum of a particle will evidently be the resultant of the momenta 
due to each of the motions a, (3, y separately ; and since the resultant of the momenta 
of all the particles will be the same in whatever way we group them together, it is 
evident that we have the following proposition : — 
“ The resultant of the momenta of a body’s particles, or the body’s momenta-system, is 
the resultant of the momenta-systems due to each of the motions a, (3, y.” 
Thus a body’s motion may be decomposed into a motion of rotation and translation. 
Hence the body’s momenta-system may be found by compounding the momenta-system 
due to the motion of rotation with that due to the motion of translation. 
Again, a motion of rotation may be decomposed into rotations about three axes. 
Hence the momenta-system of a body which rotates about a fixed point is the resultant 
of the momenta-systems respectively due to the separate motions of rotation about the 
three axes. 
55. Let us then first investigate H for a body having simply a motion of translation. 
Let V be the velocity of translation in the direction of a line AB at time t, then the 
momentum of a particle of mass m is mv in the direction of AB. Hence the momenta- 
system consists of a number of momenta parallel to one another, and proportional to the 
masses of the respective particles. Their resultant is therefore Mw at the centre of 
gravity, M being the body’s mass. Hence the momenta-system is reducible to Mv at 
the centre of gravity. Therefore H, the axis of the body’s momentum-couple about O, 
is the moment-axis about O of Mv at the centre of gravity, and is zero, if the point O 
coincides with the centre of gravity. In the latter case, since H=0, therefore D<(H) — 0, 
therefore G=0, or the moment of the external forces about any line through the centre 
of gravity is zero for a body which has simply a motion of translation. 
56. The next simplest case is that of a body rotating about a line. Take that line as 
axis of z, and suppose the body of mass M to be revolving at time t about that line with 
angular velocity ■ur. It may be easily shown in the ordinary way, that the sum of the 
moments of the momenta about the axes of x, y, z are respectively 
—Ts^lgmxz), —m%{myz), 
3x2 
