68 LECTURE IX. 



tion to be continually repeated, the effect will be greatest in a given time, 

 when the ascending weight is between two thirds and one half of the exact 

 counterpoise to the descending weight: If, however, the force were accu- 

 mulated during the action of the machine, there would be no limit to the 

 advantage of a slow motion. Thus, if we have a stream of water filling a 

 single reservoir, which is to raise a weight by means of its descent, the 

 proportion here assigned will be the best for performing the most work in 

 a given time ; but if we chose to double our machine, so that one reservoir 

 should be filled during the descent of another, it would be proper to pro- 

 portion the weights in such a manner that the whole time required for 

 filling one of the reservoirs should be occupied in the descent and the re- 

 ascent of the other. 



In all these cases, if great accuracy were required, it would be necessary 

 in the calculation to add to the mass to be moved the quantity of moveable 

 matter in the machine, reduced to a mean distance from the fulcrum or 

 centre, according to its rotatory power, in the same manner as the centre of 

 gyration is determined. But there is seldom occasion for such a degree of 

 precision. The magnitude of the pressure which is exerted on the fulcrum 

 during the motion of the connected bodies may always be determined, by 

 comparing the actual velocity of the centre of gravity with that of a body 

 descending without resistance. 



These propositions and experiments must be allowed to require an atten- 

 tive consideration from those who are engaged in practical mechanics ; and 

 it is natural to suppose that the proportions laid down may be adopted with 

 safety, and employed with success, and that we may sometimes derive 

 important advantages from their application. But on more mature consi- 

 deration, we shall find some practical reasons for caution in admitting them 

 without material alterations.* 



If a machine were constructed for raising a solid weight, and so arranged 

 as to perform its office in the shortest possible time with a given expense of 

 power, the weight would still possess, when it arrived at the place of its 

 destination, a considerable and still increasing velocity : in order that it 

 might retain its situation, it would be necessary that this velocity should 

 be destroyed ; if it were suddenly destroyed, the machinery would undergo 

 a strain which might be very injurious to it ; and if the velocity were 

 gradually diminished, the time would no longer be the same as is supposed 

 in the calculation. In the second place, the forces generally employed are 

 by no means uniformly accelerating forces, like that of gravitation, to which 

 the propositions which we have been considering are adapted : they are not 

 only less active when a certain velocity has once been attained, but they are 

 often capable of a temporary increase or diminution of intensity at pleasure. 

 We have seen the inconvenience of producing a great final velocity, on ac- 

 count of its endangering the structure of the machine : if therefore our per- 

 manent force be calculated according to the common rule, so as to be able 

 to maintain the equilibrium, and overcome the friction, the momentum or 

 inertia of the weights, when once set in motion, will be able to sustain that 



* Consult S, Gravesande's Nat. Ph. i. c. 21. Euler, Ac. Berl. 1748. Blake, 

 Ph. Tr. 1759. 



