8 Mb R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING 



11. Consider a material system at any instant of its motion. Each particle is moving 

 with a definite momentum in a definite direction, which may be resolved into components in 

 given directions in the same manner as a velocity or a force. Let this momentum be resolved 

 in the direction of a given axis OP, and its moment about that axis taken, the resolved part 

 may be called the linear momentum, and the moment the angular momentum, of the particle 

 relatively to the axis OP. Let the same be done for every particle of the system, and the 

 sums of their linear and angular momenta taken, these sums may then be called respectively 

 the linear and angular momenta of the system relatively to the axis OP. 



12. Let the linear momenta relatively to the three axes Ox, Oy, 0% be denoted by u„ Uy, 

 u^, and the corresponding angular momenta by h^, \, h^ respectively ; then it may easily be 

 shewn that the linear momentum relatively to the axis, whose direction-cosines are I, m, n, is 



lu^ + mUy + nu^, 

 and that the angular momentum relatively to the same axis is 



Ih^ + mhy + nh^. 

 The first expression will be a maximum, and equal to \u,^ + u^ + u^\i, when 



I : m : n :: Uj : Uy : u^; 

 and if this be denoted by m, it is plain that the linear momentum along any line inclined to the 

 direction of u at an angle 9 will be u co%6. Hence we may regard the whole linear momentum 

 of the system as equivalent to the single linear momentum u determinate in intensity and 

 direction. 



In like manner we may conclude that the whole angular momentum is reducible to a single 

 angular momentum h determinate in intensity and direction. 



13. Thus, just as a system of forces is reducible to a single force and a single couple, the 

 momenta of the several particles of a system are reducible to a single linear and a single 

 angular momentum, which we shall speak of as the linear and angular momenta of the system. 

 It is to be observed that the linear momentum u is independent of the origin O both as regards 

 direction and intensity, but the angular momentum h is in both respects dependent on the 

 position of O. Also it may be proved, as in the case of a system of forces, that the angular 

 momentum h remains constant, while O moves along the direction of the linear momentum u, 

 but changes, as O moves in any other direction; and finally, that its intensity will be a 

 minimum and its direction coincident with that of u, when O lies upon a certain determinate 

 line, which (from analogy) may be termed the central axis of momenta. 



14. Now let us consider the changes in the linear and angular momenta, as the time 

 changes, when the system is acted on by any forces. 



In the time dt any force P generates in the particle on which it acts the momentum Pdt, 

 and these momenta, being resolved and summed as was done above, will give rise to a linear 

 momentum Rdt in the direction of the resultant force R of the forces (P), and an angular 

 momentum Gdt relatively to the axis of the resultant couple G of the same forces. Since 

 however the internal forces consist of pairs of equal and opposite forces in the same straight 

 line, by the nature of action and reaction, the momenta produced by them will vanish in the 



