the Theory of Barker's Mill 429 



and x will be a number less than 1 : then 

 /*— x2fl 



£+m °f- 2sflh 



wherefore, by substitution, 



Pxc = ,Vxix(I-l)=Q4x^. 



If we make Jt == 1, which supposes that jf and w are infinitely 

 great, then P xv = Qh; and the effect of the machine would 

 be equal to the whole mechanic power in the quantity of water 

 Q falling from the head h. This is an unattainable limit ; but 

 the nearer x is to 1, the greater will be the moving force of 

 the machine ; which seems to be the only general rule we can 

 have to guide us in the construction of this machine. Having 

 pitched upon the most convenient value of x ; then, 



2Vgh 



v = 



a/-!-** 



a ; = 5^5 x *JF** 



effect of the machine, Px»=QAx TJ~' If j? = £, that is, 



if V be double of v 9 then the effect of the machine is § x Qh. 

 In what goes before we have taken the full velocity due to 

 the head, which is always greater than the real velocity in 

 practice. But although the real velocities are less than ac- 

 cording to theory, yet they still nearly follow the same pro- 

 portion ; that is, their squares are as the pressures, or as the 

 heights of the head. We may therefore assume 



V 2 =4g. W (£+/), 



m being a quantity less than 1, to be determined experimen- 

 tally. It is evident that this assumption does not affect the 

 equation (A). We shall now have, 



j- » = Vf_ 



