A, B. Quinb^ on the Overshot Waler- Wheel, 311 



through any fall upon an overshot water-wheel, will vary as 

 the diameter of the wheel.* 



This result being obtained, it is proper, in the next place, 

 to notice, that in practice the water does not descend in the 

 buckets entirely down to the lowest point 6, but is always 

 discharged when at an angular distance of about 30° (on the 

 wheel's circumference,) from the point h. Let this be grant- 

 ed ; and, then, 



Theorem 3. If the circumference of the wheel move with 

 the velocity due to the height of the fall above the wheel; 

 the effect which, in practice, can he derived from any quantity 

 of water, acting through any fall upon an overshot water- 

 wheel, will vary as f f of the wheel's diameter (very nearly.) 



Let?, Fig 5, be the point at which the water is discharged, 

 and tn a perpendicular drawn from t to the vertical diameter 

 ba; then, by what was proved in Theorem ], Fig. 2, the 

 effect of any quantity of water in descending upon the wheel 

 from a to t, will be to the effect of the same quantity in de- 

 scenviing from a to 6, as an to ah ; but, as an to ah, so is 43 

 to 50 (very nearly). Consequently, by what has been al- 

 ready shown, \he effect which in practice can he derived from 

 any quantity of water acting through any fall upon an over- 

 shot water-wheel, will vary as f f of the wheel's diameter. 



It now remains to determine the velocity which the circum- 

 ference of an overshot water-wheel must have, in order that 

 the effect which in practice can he derived from a given quan- 

 tity of water, shall be a maximum. 



Theorem 4. The greatest effect which in practice can he </e- 

 rived from a given quantity of water, acting through a given 

 fall upon an overshot water-wheel, is when the velocity of 

 the circumference of the wheel is the least possible. 



Put A=6F,,Fig. 5, and x=aY ; and put V=the velocity 

 due to the height aF, and v=\he velocity of the circum- 

 ference of the wheel : then, by mechanics, and from what 

 has been demonstrated, the effect which in practice can he 

 derived from a given quantity of water, acting through the 

 given fall F6, upon the wheel am6E, will vary as the quan- 



* From this demonstration, and the others here given, it will be per- 

 ceived that the whole standing theory, given in books, on the subject of 

 the overshot water-wheel, is false. 



