ON THE MOTIONS OF CONNECTED BODIES. 69 



motion equably ; and it will not be difficult to give them a sufficient mo- 

 mentum, by a greater exertion of the moving force for a short space of 

 time, at the beginning : and this is in fact the true mode of operation of 

 many machines where animal strength is employed. Other forces, for 

 instance those of wind and water, regulate themselves in some measure, at 

 least with respect to the relative velocity of the sails and the wind, or the 

 floatboards and the water ; for we may easily increase the resistance until 

 the most advantageous effect is produced. Many authors, considering the 

 pressure of a stream of water as analogous to the impulse of a number of 

 unconnected particles striking the floatboards and then ceasing to produce 

 any further effect, have inferred that the force obtained by such an impulse 

 must be as the square of the relative velocity, and that the effect of an 

 undershot wheel must be the most advantageous when its velocity is one 

 third of that of the stream : but it will hereafter appear, that this estima- 

 tion of hydraulic force is by no means accurate. If we compare the 

 greatest velocity with which a man or a horse can run or walk without 

 fatigue, to the velocity of the stream, and the actual velocity of that part of 

 the machine to which the force is applied, to the velocity of the floatboards 

 of a water wheel, the strength which can be exerted may be represented, 

 according to the experiments of some authors, by the impulse of the stream 

 as supposed to be proportional to the square of the relative velocity ; con- 

 sequently the same velocity would be most advantageous in both cases, and 

 the man or horse ought, according to these experiments, to move, when his 

 force is applied to a machine, with one third of the velocity with which he 

 could walk or run when at liberty. This, for a man, would be about a 

 mile and a half an hour ; for a horse, two or three miles : but in general 

 both men and horses appear to work most advantageously with a velocity 

 somewhat greater than this. 



Where a uniformly accelerating force, like that of gravitation, is em- 

 ployed in machines, it might often be of advantage to regulate its opera- 

 tion, so that it might act nearly in the same manner as the forces that we 

 have been considering ; at first with greater intensity, and afterwards with 

 sufficient power to sustain the equilibrium and overcome the friction only. 

 This might be done by means of a spiral barrel, like the fusee of a watch ; 

 and a similar modification has sometimes been applied by causing the 

 ascending weight, when it arrives near the place of its destination, to act 

 on a counterpoise, which resists it with a force continually increasing, by 

 the operation of a barrel of the same kind, so as to prevent the effect of 

 the shock which too rapid a motion would occasion. 



On the whole, we may conclude, that on account of the limited velocity 

 which is usually admissible in the operation of machines, a very small 

 portion of the moving force is expended in producing momentum ; the 

 velocity of 3 miles an hour would be generated in a heavy body, descend- 

 ing by its own weight, in one seventh of a second, and a very short time is 

 generally sufficient for obtaining as rapid a motion as the machine or the 

 nature of the force will allow ; and when this has been effected, the whole 

 force is employed in maintaining the equilibrium and overcoming the 

 resistance : so that the common opinion, which has probably been formed 



