On Machines in General. 221 



during the time /, and lastly, W the velocity due to the 

 height H. 



This being done, we must consider that there are two 

 sorts of forces applied to the machine, viz. those which 

 proceed from the gravity of the bodies, and those which 

 proceed from their vis inertice, or from the resistance which 

 they oppose to their change of state (note to XXX) : now 

 (XXXII) the momentum of activity consumed during the 

 time t by the first of these forces, is, with respect to the 

 whole system, M gH, or { M W ^ Let us now see what is 

 the momentum of activity consumed by the vis inertice: 

 the velocity of w being V^, and becoming the instant after- 

 wards V + ^ V, it is clear (note to XXX) that its vis inertice 



estimated in the direction of V, is m d V, or rather m . 



therefore (XXX) the momentum of activity, exercised by 



this force during d t, is ?w Vdt,ori7iVdV : therefore the 



d t 



momentum of activity, consumed bv this vis inertice du- 

 ring the time /, is s mW dY , or, by integrating and com- 

 pleting the integral, \ mV^ — ^mK' : thcreiorethe momen- 

 tum of activity, consumed at the same time by the vis 

 inerticB of all the bodies of the system, will be y 5 to V^ — i 

 f m K- : now this vis inertice is a resisting force, since it is 

 by it that bodies resist their change of state : and the weight 

 is here a soliciting force, since the centre of gravity is 

 supposed to descend : thus, by the proposition of this co- 

 rollary, we should have M W^ = sniY'- — s m K', or s m 

 W = smK' + MW^; i.e. 



In a machine with weights, the movement of which chancres 

 by insensible degrees, the sum of the active forces of the sy. 

 stem is, after any given time, equal to the sum of the ini' 

 tial active forces, plus the sum of active force which would 

 take place if all the bodies of the system were animated xvith 

 a common velocity, equal to tiiat which is owing to the height 

 from which the centre of gravity of the system has descended. 



XLIi. If the movement of the machine l)e uniibru), we 

 shall continually have V = K, and iheicfure \V= = o, or 

 H = : this teaches us that 



