•.fJL6 On Machines in General. 



, positions; but that there is no fixed point or any obstacle; 

 . ,the equation (F) gives us the solution of this problem on 

 ^attributing successively to 7i different values and directions. 

 |,, 1st. As the velocities 2/. are not subjected to any condi« 

 ^lion, unless the movement of the system in virtue of which 

 .Ithe corpuscles m have these velocities be geometrical, it is 

 .jCvident that we can at first suppose all of them equal and pa- 

 ,,Tfillel to one given line : then u being constant, or the same 

 ij^ith respect to all the points of the system, t]»e equation (F) 

 .will be reduced to s m U cosine 2: = 0; which informs us 

 that the sum of the forces lost by the reciprocal action of the 

 bodies in the aibitrary sense of u is null, and that conse- 

 quently that which remains is the same as if each body had 

 been free; this is a well-known principle, 



Sdly. Let us now imagine that we make the whole system 

 turn round a given axis, so that each of the points will de- 

 scribe a circumference round this axis, and in a plane which 

 shall be perpendicular to it ; this movement is visibly geo- 

 metrical; therefore the equation (F) takes place: hut then 

 on calling R the distance from vi to the axis, it is clear that 

 we have 21 = AR'', A being the same for all the points ; 

 therefore the equation (F) is reduced to 5 w R U cosine 

 z — 0; that is to say, that the sum of the momenta of the 

 forces lost by the reciprocal action relatively to any axis is 

 null ; this is another well-known principle, 



3dly. We might also attribute to u other values ; but this 

 would be useless, and might lead to equations already pon- 

 tained in the preceding; for we know that the latter are 

 sufficient for resolving the question, or at least fqr reducing 

 it to a matter cf pure geometry. 



First Remark, 

 XVI IT. The object we propose by giving a geometrical 

 movement is to change the state of the system, without al- 

 tering however the reciprocal action of the bodies which 

 compose it, in order i hereby to procure relations between 

 these exercised and unknown forces and the arbitrary velo- 

 cities which bodies assume in virtue of these different geo- 

 metrical 



