S 1 8 On Machines in General, 



y the cm gle formed by V and u, d t the element of the time ; tee 

 shall have these two equations: 



smVpd /cosineR — s wVfl^ V=0: 

 smup d t cosine r— s mud (V cos y.) = 0. 

 Demonstration. — In the first place, pdt cos R is visibly 

 the velocity which the vis motrix p would have impressed 

 ujx)n w in the direction of V, if this body had been free .- 

 besides, dY \s the velocity which it would in reality receive 

 in the same direction ; therefore pdt cosine R — (i V is 

 the velocity lost by 7n in the direction of V, in virtue of the 

 reciprocal action of the bodies : it is therefore this quantity 

 that we must put for U cos. Z in the fundamental equation 

 (E), which becomes by this substitution s mV p d t cosine. 

 R ~ 3' m V dV = o', being the first of the two equations 

 which we had to demonstrate. 



Secondly, pdt cosine r is the velocity which the vis mo- 

 Irix p would liave impressed upon m in the directioDj of 7/, 

 if this body had been free ; besides, V cosine y being the 

 velocity of m in the direction of ^^, d (V cosine y) is the 

 quantity which this velocity estimated in the same direction 

 auiiments : therefore pdt cosine r -^ d (V cosine y) is the 

 velocity lost by m in the direction of 2i, in virtue of the re- 

 ciprocal action of the bodies : it is therefore this quantity 

 which we must put for U cosine z in the second equation (F), 

 which becomes by this substitution s mup d t cosine r — 

 smu d {Si cosine y) = 0, which is the second of the twa 

 equations we had to demonstrate. 



These equations are therefore nothing else than the fun- 

 damental equations (E) and (F) applied to the case where 

 the movement changes by insensible degrees, and therefore- 

 thev contain all the laws of this .movement : we may re- 

 mark also, that the first of th^se two equations is only a par- 

 ticular case of the second, for the same reason that the 

 equation (E), whence it is extracted, is contained in that 

 (F) whence the second is extracted. Bui this first equation 

 s mV p d t cosine \{ -- s mV dV = deserves particular 

 attention; because it contains the famous principle of the 

 preseiration of active forces in a system of hard bodies the 

 movement of which chanirei bv insensible degrees : thus ; 



Let 



