specific names to all such functions of simple elements^ ^c. 29 
tion will make it appear that no alteration can arise from 
deflecting the forni 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 by ~, 
h -jj = the pressure sustained, or the virtual head, and 
) X .r=the efficiency applied to the machine, in relation 
however, not to the expenditure of fluid corresponding to the 
height (A), but to an expenditure greater in the proportion of 
to 1. But a portion of this efficiency must, of 
necessity, be expended in giving the rotary velocity (jc) to 
the quantity of fluid issuing z=\^i —j . Now the height 
due to X IS , consequently v 1 -h y,— y. x =: the ex¬ 
penditure of efficiency in producing the rotary motion. And 
— vi + X J X = the efficiency applied 
to the machine capable of producing duty. 
Now substitute for jc, (the velocity due toh) [2 \/hi) mul¬ 
tiplied by an arbitrary quantity y; then 1/1 (or the 
quantity of fluid issuing) = 1/1 -j- and (y -f y3 _ 
xy^) X 2/12 = the efficiency; and dividing by \/i -|-y ^ 
1/1 +y*xy^x2A2’/2' = the efficiency that would be applied 
if no more fluid issued than what is due to the actual head [h). 
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 y^^ths to a.25 ; after which the duty 
becomes negative. 
