and on the Maximum Effect of Machines. 421 



express the reaction, which being multiplied by v, the space 

 through which it acts in a second, gives 2 av (h+ -j-) for 



the total moving force of the arm in a second. But a part of 

 this moving force is expended in producing the rotatory mo- 

 tion of the water, and in raising it to the height — . For, if 



we suppose a perpendicular tube CP to rise from the arm at 

 C, the surface of the water in that tube would stand at P, PR 



being = — . Now if instead of letting the water escape at 



C, it be allowed to flow over the perpendicular tube at P, and 

 fill another similar perpendicular tube adjoining it, and issue 

 from an orifice at the bottom of that tube, the effect must be 

 the same as if it issued at C, and a moving force must be ex- 

 pended at C, sufficient to generate the velocity v, in the water 

 which passes, and also to raise it from R to P. 



The pressure at C being equal to the weight of a column of 



water whose height is h + .-— , (that is = AB + PR), the ve- 



locity with which the water issues will be 4 / 4g (h + -~-) 



or »J 4>gh + vK Let V express that velocity, then «V will 

 express the quantity of water which passes in a second ; and 



2 a V — will express the moving force necessary to generate 



the velocity v, in that quantity of water, and to raise it from 

 R to P. That quantity of moving force being deducted from 



the total moving force of the arm, leaves 2av(h + — \ 



— 2aY — for the effective moving force of the arm in a second. 



That this is the effective moving force, may be shown also 

 in another manner, as follows : 



The absolute velocity of the water after it has left the ma- 

 chine will be V— v, and ; ■ will be the head which would 

 produce that velocity; which being multiplied by «V, the 

 quantity of water delivered in a second, gives a V ^ for 



the moving force which remains with the water after it has left 

 the machine. 



If that be deducted from aVh, the whole moving force of 



the water, there will remain a\ h — aV ~ v * for the effective 



moving force, which will be found to be equal to 2av(h + -^-) 



The 



2 a V-^, the effective moving force stated above. 



