VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE. 9 



summation over the whole system ; we may therefore regard R and G as the resultant fprce 

 and resultant couple of the external forces. Then the linear momentum u along the line OU 

 must be compounded with the linear momentum Rdt in the line OR in order to obtain its value 

 at the time t + dt: and in like manner the angular momentum h relatively to the axis OH must 

 be compounded with the angular momentum Gdt relatively to the axis OG. 



15. Hence the method of the previous section applies to momenta of both kinds, replacing 

 / in one case by R and in the other case by G. Thus the equations {B) give us 



du dd) 



— = RcosRU, u-^= R sin RU, 



dt dt 



where d(b is the arc through which U moves towards R in the time dt : and 



— = G cos GH, A -/^ = G sin GH, 



dt dt 



where d^ is the arc through which H moves towards G in the time dt. 



Also for fixed rectangular axes, with respect to which the components of R and G are 

 X, Y, Z and L, M, N respectively, it is plain from the above reasoning that we should have 



dt ~ ' dt ~ ' dt ~ ' 



dh^ d\ dh^ 



dt dt dt 



which are really the six fundamental equations of motion of our works on Dynamics. 



For rectangular axes moveable about 0, the equations {E) of the last section furnish 

 two sets of three equations, of which the types are 



<iM^ „ „ „ 



-^ = -y + up., - u^Q,^, 



16. If the system be acted on by no external forces, it follows that both u and k 

 are constant in intensity and invariable in direction. This result might by analogy be 

 named the principle of the Conservation of Momentum. 



This principle, as applied to linear momentum, is obviously equivalent to the prin- 

 ciple of the conservation of motion of the centre of gravity : as applied to angular 

 momentum, the constancy of direction of the axis of h and therefore of a plane perpen- 

 dicular to it shews that there is an invariable axis or plane, while the constancy of its 

 intensity and therefore of its resolved part in any fixed direction is equivalent to the asser- 

 tion of the truth of the principle of the conservation of areas for any fixed axis. 



It may also be noted that there is an infinite number of invariable axes, and that, 

 if the origin be taken on the central axis of momenta, the corresponding invariable 

 axis will coincide with the central axis, and the angular momentum about it will then be 

 Vol. X. Paet I. 2 



