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 

 ? t ?7&amp;gt; ? \ P&amp;gt; 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 



- - 



- 



...(!)*. 



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

 at time t angles whose cosines are 



(1, rSt, -q&t), (-rSt, 1, ptt), (qtt, - pSt, 1), 



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

 axis of x, 



+ gf = f + n . rSt - . qSt + XSt. 



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

 membering that has been displaced through spaces u&t, v&t, 

 parallel to the axes, we find 



\ + &V. = \ 4- 77 . wSt - f . v8t + 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 



^ const.,) } 



? = const. J 



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

 resultants of the impulse are constant. 



* Cf. Hay ward, &quot; On a Direct Method of Estimating Velocities, Accelerations, 

 and all similar Quantities, with respect to Axes moveable in any manner in space.&quot; 

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



