122 Mr. W. M. Buchanan's Theory of the Reaction Water-Wheel. 



in those of M. Bidone, the ratio from difference of size of orifice (cir- 

 cular of 0'02 to 0-036 metre's diameter,) varied from—— to 5-555! 



These seeming anomalies are explicable by reference to a principle 

 which will be immediately adverted to ; but in the meantime it is neces- 

 sary to inquire into the circumstances by which the actual mean head 

 of ten feet was reduced, when measured by the velocity of efflux at 

 the orifices, to 8*5682 feet, showing a loss of 1*4318 feet of head- 

 pressure incurred between the reservoir and the orifices of the machine. 

 It was easy to perceive that this loss did not result from a single 

 cause, but expressed the conjoined effect of several influences which it 

 was necessary to determine individually. In the first it was obvious 

 that there would be a certain amount of head-pressure absorbed by 

 the friction of the water in passing through the supply-pipe. This was 

 regarded as a known quantity, which could be represented in character 

 and amount by 



2/. 1 L. *' 

 A, ig 



in which C denotes the internal perimeter, A / the cross-sectional area, and 



L the length of the pipe : u the velocity with which the water descends 



through it, and/ an empirical coefficient = *0035. If therefore S denote 



the sum of the areas of the orifices, V the velocity of efflux, and D the 



diameter of the pipe (all in feet), this equation may be put under the 



form 



J ' D ' A, 2 ' 2g - * 2g 

 Another small but permanent influence, tending to diminish the pres- 

 sure, is the acceleration experienced by the water in passing from the 

 supply-pipe into the interior of the machine through the neck formed by 

 the mouth-piece and central opening, and which are commonly less in 

 diameter than the supply-pipe. This was likewise known from established 

 data to be represented in form by 



A, 2 V m / 2g 2g 



in which A„ is the area of the central opening, and u the velocity of the 

 water passing through it : m a coefficient, determined by a very extensive 

 scries of experiments directed exclusively to that object to be = -9378. 



A third, though very small loss of pressure would obviously result from 

 the resistance encountered by the water in traversing the arms of the 

 machine. The effect of this resistance, were the channels uniformly 

 contracted from their base outwards, would be represented by 



8/s 2 . p. /* <Ld x 



2 £*/0 A /// 

 in which C / and k. ln are respectively the transverse perimeter and area of 



