specific names to all such functions of simple elements , &c. 2 9 
tion will make it appear that no alteration can arise from 
deflecting the form of the arm out of a straight line. 
Then, since the pressure exerted by the actual head is ex- 
pressed by its height ( h ), and the pressure from centrifugal 
force hy —■ , 
h ~ = the pressure sustained, or the virtual head, and 
| A _j_ ■— J x x = the efficiency applied to the machine, in relation 
however, not to the expenditure of fluid corresponding to the 
height ( h ) , but to an expenditure greater in the proportion of 
^,2 \ m • e 
V(i + -py) to 1. But a portion of this efficiency must, of 
necessity, be expended in giving the rotary velocity ( x ) to 
the quantity of fluid issuing = V 1 + . Now the height 
due to x is ~ , consequently V 1 + x x x = the ex- 
penditure of efficiency in producing the rotary motion. And 
(ft + -p — V 1 + x -~) x x = the efficiency applied 
to the machine capable of producing duty. 
Now substitute for x, (the velocity due to h) (vVhl) mul- 
tiplied by an arbitrary quantity y ; then V 1 + ( or the 
quantity of fluid issuing) = y/\ -j" y 2 and (y + y 3 — Vi + y* 
xy 3 ) x 2 A 2 / 2 = the efficiency ; and dividing by yT -j-y 2 
Si+y*xy a xaHli = the efficiency that would be applied 
if no more fluid issued than what is due to the actual head (ft). 
The following Table exhibits the value of both these func- 
tions, taking (y) from 0 (where the duty must obviously be 
nothing) by steps of Troths to ,1.25 ; after which the duty 
becomes negative. 
