430 Mr. Ivory's Notes relating to the Theory of Barker's Mill. 



= 2/ = 2h x - — 

 <j l — ' 



Q,g J 1 — m i* 



t» tt k f 2 / V _ \ ~ , 2m(x—x*) 



P x v = aV x — x( 1 ) = Q# x -^ — -r* 



Now it is obvious that the expression ' ~Z is susceptible of 

 a maximum ; and, by applying the usual rule, the value of x 

 answering to the maximum will be found ; viz. x = 1 ~v 1 ~^ 



m 



This expression shows that, whatever fraction of unit m stands 

 for, x is contained between the limits \ and 1* ; so that, when 

 the machine works to the greatest advantage, v is greater than 

 ^ V and less than V. With a given value of m, the rules for 

 constructing the machine so as to work to the greatest effect, 



are contained in the following formulas, in which n = — , viz. 



i?= 2 s/ g h x .- 



° sy/n-x* 



V=2 4/jA X 



a/ n— x* 



Q 



«=77= * */ n-x\ 



effect of the machine, P x v = Q h x (l — V 1 — m) f. It now 

 appears that, m falling short of 1, the effect of the machine 

 decreases rapidly. If m = -8, then Pxd=Q^X 0*5528, 

 or P x t> = | x Qh 9 nearly. 



If m = |, that is, if half the pressure were lost in forcing 

 the water through the tubes, the effect of the machine would 

 be reduced to Qh x 0*293, or nearly T 3 n X Q h. 



* i" 



* ™ = 2 + 8 + 16 + 128 ^ 



when m = 0, * = \ ; when m = 1, x - 1. 



+ 1- jyT^m 

 i -j _- 2L , 



m 



mx % = 2 a* — 1, 

 1 — WJ7 9 = 2(1 — .r) 



2«(i-i') 2m(i-^) . ,- 



LXV. Ow 



