312 A. B. ^uiuby on the Overshot IValer-lVhetl. 



tity (V-?;)2 4-Af(A — x); which will manifestly be a maxi- 

 mum when V is the least possible.^ 



Theorem 3. If the circumference of an overshot wheel 

 move with the velocity due to the height of the fall above the 

 wheel, the product of the velocity and circumference of the 

 wheel will be the greatest when the diameter of the wheel is 

 f the height of the whole fall. 



Put A = BF, Fig. I. and x^^hYi then the circumference 

 will be 3J4159, &;c. [h — x^]-, and the product of the velo- 

 city and circumference will be \/'2gx^ X3. l4159,&to.(A- x^); 

 which is to be a maximum ; or, by rejecting constant factors 

 x{h-x^)=hx—x^=max. Wherefore, Ax — 3x2a;=0 ; or, 

 x^=^h; and A— x^=|/t.t 



Theorem 6. If the circumference of an overshot wheel 

 move with the velocity due to the height of the fall above the 

 wheel, the wheel will have its greatest angular motion, or will 

 perform the most revolutions in a given time, when its diam- 

 eter is the least possible. 



Retaining the same notation, we have for the angular motion 



of the wheel q , !. , . n ~5 — Tt :rT"= ^ . ?. r- »o X 



3.l4l59,&;c.(A-x2) ~3.14l59&c. ^h-x^ ' 

 which will obviously be greatest when the quamity A — x-, 

 (the diameter of the wheel,) is the least possible. 



Note. The author deems it proper to state, that all the 

 preceding demonstrations were made out in January last ; 

 but for certain causes were not arranged and offered for pub- 

 lication until the present time. 



New- York, Dec. 28, 1824. 



Remark by the Editor. — Mr. Quinby, in a letter to the 

 Editor, dated Feb. 12, U25, promises to communicate a pa- 

 per, for a subsequent number of this Journal, applying the 

 above theory to the " Fitch-back" and ''Breast Wheels." 



*Dr. Gregory, in his Mechanics, states that,"" In an overshot wheel, 

 tke machine will have its g-reatest perfection," or, as he expresses it in 

 a subsequent edition, " will have its greatest rotary velocity, when the 

 diameter of the wheel is |- of the height of the Avater above the lowest 

 point of the wheel." Both these enunciations are obviously false. 



t This demonstration exhibits what, in Dr. Gregory's Mechanics, is 

 given to express the greatest perfection of the machine ; or, as it is in 

 the last edition of liis work, the greatest rotary velocity (meaning, it 

 is presumed, angular velocity) of the wheel. But it is plain that the 

 quantity exhibited in this demonstration does not expi-ess the greatest 

 perfection of the machine ; nor the greatest rotary velocity of the 

 wheel. 



