304 On Machines in General, 



quantity in which this halt' amount will be inrreased by the 

 change (XLV). 



Finally, supposinc; we have anv system "of bodies, that 

 these bodies attract each other, on account of any func- 

 tion of their distances; even supposing, if we please, that 

 this law is not the same with res|icct to all the parts of the 

 system, i. e. that this attraction follows any law we please, 

 (providing that, between two given bodies, it only varies 

 when the distance of these bodies in itself varies,) and it he 

 required to make the system pass from any given position 

 to another : this being done, wlialcver be the path that we 

 ■wish each of the bodies to take, in order to attain this ob- 

 ject, whether we put all these bodies in motion at once, or 

 the one after the other, whether we conduct them fiom one 

 -place to another by a rectilinear or curvilinear motion, and 

 varied in any manner (providing no shock nor rapid change 

 occur) ; lastly, whether we employ any kind of machines 

 ■whatever, even by a spring, providing that in this case we 

 ultimately replace the springs in the same state of tension 

 in which they were at the first moment, the mumenlmn of 

 aclivity which they will have to consume, in order to pro- 

 duce this effect, the external agents em[)loyed to move this 

 system, will always be ihe same, supposing the system to be 

 at rest at the first instant of the n)ovemcnt, and at the last 

 also. 



And if, besides all this, it be necessary to produce in the 

 system any criven movement, or if it be already in motion at 

 the first moment ; and if it be requisite to modify or change 

 this movement, the momentum of activity which the exter- 

 nal agents will have to consume will be equal to that which 

 it would be necessary to consume if it were merely requisite 

 to change the position of the system, without impressing 

 any motion upon it (?. e. considered as at rest at the first 

 and last instants,) plus the half of the quantity by which 

 we must augment the sum of the active forces. 



It is of very little importance therefore, as to the expen- 

 diture or momentum of activity to be consumed, that the 

 forces employed arc great or small, that they eniploy such 



and 



