107109.] IMPULSE. 123 



external impressed forces this moment of momentum is constant 

 throughout the motion. Hence the moment of the impulse about 

 any line through P is constant. Since in this argument P may 

 be any point within a finite distance of the solid, it follows that 

 the moment of the impulse about any line whatever is constant. 

 This cannot be the case unless the impulse is itself constant in 

 every respect. 



We see in the same way that if any external impressed forces 

 act on the solid, the moment of the impulse about any line is 

 increasing at any instant at a rate equal to the moment of these 

 forces about the same line. 



The above are somewhat modified proofs of theorems first 

 given by Thomson*. It should be noticed that the reasoning still 

 holds when the single solid is replaced by a group of solids, which 

 may moreover (if of invariable volume) be flexible instead of rigid, 

 and even when these solids are replaced by portions of fluid moving 

 rotation ally. 



109. The impulse then varies in consequence of the action 

 of the external impressed forces in exactly the same way as the 

 momentum of any ordinary dynamical system does. To express 

 this result analytically let 77, f ; X, p, v denote the components 

 of the force- and couple-constituents of the impulse ; and let 

 X, F, Z\ L, M, N designate in the same manner the system of 

 external impressed forces. The whole variation of f, 77, f, &c., 

 due partly to the motion of the axes to which these quantities 

 are referred, and partly to the action of the forces X, Y, Z, &c., 

 is then given by the formulae f: 



JL = rrj _ ^ + X, C -j- = WTJ - v + rp - qv + L, 



(20). 



at, ui/ 



-ji = &amp;lt;-pn + z &amp;gt; 7r f = l % 



* On Vortex Motion. 



t See Hayward, Camb. Trans. Vol. x. 



