126 Mil. W. M. Buchanan's Theory of the Reaction Water-Wheel. 



with exactness, the motive force derivable from any given head will be 

 known. Generally the value of that head will be expressed by 



2# 2$r 



and, therefore if, for simplicity, we put * + /3 -f y + &, + e = K, we 

 shall have 



- V l + K 



This being the velocity of efflux, the expenditure of water in a second 

 will be indicated by 



2gR 



«»vs 



cubic feet. 



K 



S being the sum of the areas of the two orifices ; and if w be the weight 

 of a cubic foot of water, and P the weight which would be sufficient to 

 balance the reaction due to that discharge, we shall have 



P=JL S _X2cH 

 l + K 



in which c is the coefficient of reaction depending upon H and S, and 

 which in the experiments before referred to, was found to give 

 2c = 21105. 



Theory of the Action of the Machine. — From what is here stated, it is 

 clear that if a weight p, less than P, be applied to the machine, a certain 

 amount of the reaction will remain to generate velocity. But when 

 motion is induced, a new order of things will arise. Centrifugal force 

 will be immediately produced in the water occupying the arms of the 

 machine, and the pressure at the orifices being thereby increased, the expen- 

 diture of water will be correspondingly augmented. 



A common expression for the centrifugal force referred to a unit of 

 mass revolving in a circle is tf% when 9 =z the angular velocity of the 

 body, and ^ its distance from the centre of rotation. Now, if the unit of 

 mass advance in the direction of the radius outwards through the element 

 of space dx, in the time dt, the force developed in that direction by the 

 centrifugal force, will be &*xdx; and if this be integrated for the space 

 R — r, the length of the arm of the machine, we shall have 



%x<k = i4 2 (ft 2 --r 2 ) 

 r 

 And putting v for the absolute velocity of the machine in feet per second, 



at the distance R, from the axis of rotation, we have 6 = — -. Substituting 



R 

 this value of 6 in the equation just found, and taking the weight of the 

 element of fluid = 1, we have as the increment of pressure due to the 

 velocity v, 



