and on the Maximum Effect of Machines. 423 



be PR, which is also the head due to v, the velocity of the 

 orifice. We shall therefore have V = v; and if CP represent 

 the total moving force necessary to raise the water from C to 

 P, CR = AB will represent that part of it which is expended 

 in producing change of figure. The greatest velocity, there- 

 fore, that the orifice, when the machine meets with no resist- 

 ance, can acquire, will be V 4gx4A. 



When the velocity of the orifice is less than that, V will be 

 greater than v; and V— v, the absolute velocity of the water 



after it has left the machine, will be *J # 8 (4<gh + v*)— v. The 

 head or the moving force expended in producing that velocity 



winbeM v^±gg>'. 



The moving force expended in producing change of figure 



will be as '2 (h + ^— )• Now when the sum of these two 



\ 4g / 



^2 



quantities, or — — i-j- - — + '2( h + — j, is a minimum, 



we shall find v = V c 2gh{^~b' — l) = 6-3056 V~h for the 

 velocity of the orifice when the machine produces a maximum 

 of effect; and in that case the above sum becomes = '4472^. 



We shall therefore have h — # 4472^ = *5528h for the maxi- 

 mum of effect, supposing h to represent the whole moving 

 force of a given quantity of water descending from A to B. 

 This effect is considerably greater than that which the same 

 quantity of water would produce if applied to an undershot 

 water-wheel, but less than that which it would produce if pro- 

 perly applied to an overshot water-wheel. 



Respecting the maximum of effect produced by machines, 

 I wish to observe, that in the actual construction of machines 

 it is necessary to aim at a maximum quite different from that 

 which is usually proposed in books on the theory of mechanics. 

 This will perhaps be best ex- 

 plained by examining the sim- 

 ple case where a given weight 

 P, (fig. 3) connected with an- 

 other W, by a string passing 

 over the pulley F, descends 

 vertically and raises W, with- 

 out friction, from the hori- 

 zontal line AC along the in- 

 clined plane AB. If we 

 make AB : BC : : 2 W : P, W will be raised to B in the least 



time ; 



