116-117] IMPULSE OF THE MOTION. 171 



impulse ; and let X, Y, Z, L, M, N designate in the same manner 

 the system of extraneous forces. The whole variation of 

 f *)> ? \ P> v, due partly to the motion of the axes to which 

 these quantities are referred, and partly to the action of the 

 extraneous forces, is then given by the formulae 



-?=rri 





For at time t + St the moving axes make with their positions 

 at time t angles whose cosines are 



(1, r&t, -qSt), (-rSt, I t p&t), (q&t, -pit, 1), 



respectively. Hence, resolving parallel to the new position of the 

 axis of x, 



% + g = f + v . r8t - f . q$t + XSt. 



Again, taking moments about the new position of Ox, and re- 

 membering that has been displaced through spaces uSt, v8t, wbt 

 parallel to the axes, we find 



= X -}- 77 . wSt .v$t + p. rSt v . qSt + LSt. 



These, with the similar results which can be written down from 

 symmetry, give the equations (1). 



When no extraneous forces act, we verify at once that these 

 equations have the integrals 



? 2 +7/ 2 + ^ = const, j , 2 , 



\% + ft?) + v = const, j ' 



which express that the magnitudes of the force- and couple- 

 resultants of the impulse are constant. 



* Cf. Hay ward, " On a Direct Method of Estimating Velocities, Accelerations, 

 and all similar Quantities, with respect to Axes moveable in any manner in space." 

 Camb. Trans., t. x. (1856). 



