10 mk r. b. hayward, on a direct method of estimating 



a minimum: also that for any other position of the origin the direction of the invariable 

 axis and the intensity of the momentum about it will depend upon the position of the line, 

 parallel to the central axis, in which the origin lies, just as in the corresponding propositions 

 relative to couples, 



17. Any one of the different sets of equations in § (15) may be used to determine 

 completely u and h, when the forces are given or vice versa. It is to be observed that 

 the equations involving h, refer either to -a fixed origin, or to an origin, whose motion is 

 always in the instantaneous direction of u the linear momentum, for, as we saw, a change of 

 the origin in this direction does not produce a change in h, as its change in any other 

 direction does. It would be easy to introduce terms depending on the motion of the origin ; 

 in the last set of equations, for instance, if a^apO^ denote the linear velocities of the origin 

 in the directions of the axes, the equation for h^ becomes 



—r- = L + hyQ^ — h^Qj, + u^z — uja.y. 



The equations involving u, are entirely independent of the origin, and will there- 

 fore not be affected, however the origin be supposed to move. 



18. It appears then that the linear and angular momenta are determined solely by the 

 external forces acting on the system, and not on the system itself otherwise than the forces 

 themselves depend on it : in fact, they are simply the accumulated effects of the forces and 

 the initial momenta. To proceed to the determination of the actual motion of the system 

 from these momenta, the system must be particularised, and as one system may differ from 

 another both as to the quantity of matter included in it, and as to its arrangement, 

 we may consider separately how much farther particularisation in either respect will enable 

 us to carry our results. 



19. If the quantity of matter or mass of the whole system be given, it is well known 

 that the linear momentum of the system is that of its whole mass collected at its centre 



of gravity, so that, M denoting this mass, the velocity of the centre of gravity is — in 



M 



the direction of the linear momentum : thus the motion of a certain point definitely related 

 to the system is obtained, and this is usually regarded as defining its motion of trans- 

 lation. For any other point definitely related to the system, the motion will in general 

 depend also on h and the arrangement of its matter. 



20. If then the translation of the system be referred to its centre of gravity, its 

 motion about the centre of gravity will depend solely on h and the arrangement of its mass ; 

 for the direction of motion of the centre of gravity being that of the linear momentum, h 

 referred to that point as origin will be independent of u. Now the arrangement of a system 

 of matter may be either permanent or variable. If the former, it is spoken of as a body 



