specific names to all such functions of simple elements, &c. 31 
employed with advantage. The water wheel, whenever the 
fall does not exceed forty or fifty feet ; and the pressure 
engine, provided with an air vessel, in all cases where the fall 
is greater, receive an efficiency almost equal to the whole 
expended, instead of one half ; and in practice, the duty will 
reach three quarters. The wheel moreover possesses a self- 
regulating power, sufficiently accurate for the present illustra- 
tion, and somewhat analogous to the isochronism of the pen- 
dulum ; that is, the actual weight of water on the wheel, and its 
velocity of rotation, must always be reciprocal to each other ; 
thus maintaining a given efficiency, independent of velocity. 
The received efficiency of a water wheel being represented 
by the pounds of water passing over it x by the fall in feet, 
less the height due to the velocity with which the periphery 
moves in its rotation. But in the recoil engine, the moving 
power is expending at the rate due to the height even when 
the machine is actually standing still ; and when it moves 
with a velocity exceeding that due to the length of the head, 
in the proportion of about 128:1 (where the duty is again 
nothing ) the moving power will be expended at the increased 
rate of V 1 + ( 1.28)* : 1. 
Here too may be easily shown the utter impossibility of 
executing a plan, which would not indeed deserve any notice, 
were it not that two ingenious practical engineers have within 
these few years, although at different times, and independently 
of each other, incurred large expense, and employed much 
pains in attempting to apply steam on the principle of recoil. 
It is obvious that one of the factors, namely force (or the pres- 
sure of recoil), must in this case be extremely small, since an 
aperture of one inch square will discharge about 3 imperial 
