On Machines in Genei'aL 319 



Let us first call d s ilie element of the curve described by 

 the corpuscle m during d t ; this being done, we shall have 



V d t — d s f and therefore the preceding equation assumes 

 this form smpd s_ cosine R — 57?iVt?V = 0. Now let us 

 suppose for a moment that the curve described by m is an 

 inflexible line, ihat m is a. moveable grain interwoven with 

 this curve, that it traverses it freely, i, e. without being 

 pressed by the re-actions of the other parts of the system, 

 that it experiences at each point of this curve the same 

 vis motrix as that with which it was animated in the first 

 case; and that, finally, in this first case the initial velocity 

 of w is K, while in the second it will be null at the first 

 instant, and V'' after an indeterminate time / ; this being 

 done, by integrating the preceding equation, in order to 

 have the state of the system at the end of the time /, we 

 shall have for the first case s' s m p d s cosine R — 5' 5 m 



V c? V = 0, s' designating the sign of integration relative 

 to the duration of the movement, while s is the sign of in- 

 tegration relative to the figure of the system : no\v_, s' s m 



s mV^ . 



V fi V = • therefore the equation may be placed ia 



jt 



this form s' s m p d s cosine R — 5 ?» V* -f C = ; C 



being a constant added to con)plete the integral : in order to 



determine it, we shall observe that at the first instant we 



have V = K and s' s m p d s cosine R = ; therefore 



smYJ- 

 C = — - — 5 therefore Is' s m p d s cosine R — 5 tw V* 



s mYi} = : by the same reasons we have for the second 

 case 2 s' s m p d s cosine R — 5 ?« V' * = 0, without a con- 

 stant, because we suppose V' as null at the first instant: 

 taking away therefore this e<juation from the preceding one, 

 reducing and transposing, we have s m V- •=■ s m }L- -^ s m 

 V'*; that is to say, in any system of hard bodies the mtve- 

 ment of which changes by insensible degrees, the stnn of the 

 active forces at the end of arnj given time is equal to the sum 

 of the initial active forces, plus the sum of the active forces 

 ivhich would take place if each moveable particle had for its 

 velocity ihat which it would have acquired by freely traversing 

 iJie curve it had described, and supposing bendes that it had 



bun 



