424 Mr. Ewart on the Reaction of effluent Water, fyc. 



time*; and upon this principle, the maximum of effect in 

 machines is usually demonstrated in theory. In practice, how- 

 ever, the object is not merely to raise W to B in the least 

 time, but to raise it with the least expenditure of moving force. 

 When it is raised in the least time, P must descend through 

 a space = AB, but when it is raised with the least moving 

 force, P descends through a space = JAB only. For, if we 

 make BD= JAB, and let W ascend along any concave sur- 

 face DEB, of which BD is the chord, it will be raised to B 

 by the descent of P through a space = BD, and it will be at 

 rest when it arrives at B. This is .so obvious, that it would 

 be superfluous to give a demonstration of it. It appears then, 

 that twice the quantity of moving force which is absolutely 

 necessary to raise W to B, must be expended if it is to be 

 raised by P in the least time. To determine the curve by 

 which W will ascend from D to B in the least time, is an in- 

 tricate problem, and I do not know that it has ever been 

 solved f; but a practical approximation to it in any particular 

 case may be easily found. A well constructed steam-engine 

 for raising water exhibits in every stroke a practical example 

 of the same problem. At the commencement of the stroke, 

 a very great pressure of steam is thrown upon the piston, and 

 this pressure is gradually diminished, so that at the end of the 

 stroke there is a considerable preponderance in the opposite 

 direction. In consequence of this regulated pressure of the 

 steam, the motion of the machine resembles the uniform vibra- 

 tions of a pendulum, and the moving force of the steam is 

 applied to the greatest advantage. 



By proceeding on the principle that when W is raised to B 

 in the least time, the maximum of effect is produced, many 

 erroneous conclusions have been drawn respecting the proper 

 construction of machines. It is laid down for example, on 

 this principle, that " In an overshot water-wheel, the machine 

 will be in its greatest perfection, when the diameter of the 

 wheel is two-thirds of the height of the water above the lowest 

 point of the wheel J." But it is very well known that there 

 would be lost, by that construction, nearly one-third of the 

 moving force of the water, which is saved by making the wheel 

 one-half larger in diameter, and by making its velocity much 

 less than what is required by the above rule. 



* If the ascent be made in the least possible time, W must ascend not 

 along the plane AB, but along a concave surface AGB. 



f This difficult problem, we understand, has lately been solved by the 

 Rev. E. Sibson of Ashton, in Makerfield, Lancashire, and the solution will 

 appear in the next volume of the Memoirs of the Lit. and Phil. Society of 

 Manchester. — Edit. 



X Gregory's Mechanics, vol. i. p. 447. 



Notes 



